Deep Learning Course for Beginners

Key Takeaways

this deep learning course is designed to take you from beginner to proficient in deep learning IU sing created this course he's an experienced data scientist and popular course creator aush will teach you the fundamental concepts architectures and applications of deep learning in a clear and practical way so get ready to build train and deploy models that can tackle real world problems across various Industries what if I tell you that there's a deep learning course that teaches you deep learning from very scratch to a core level so what I really mean by scratch is teaching you the core and the Crux of the mathematics which is required for deep learning like linear algebra single variable calculus and much more and not only this we give you detailed lecture notes along with the Practical assignments on data Wars for absolutely free so that you can follow through this course and become the master hi this is aayush I'm co-founder of second brain laps and in past worked as a lead data scientist at triplate a UK based esteemed organizations working on large scale career economy product as well as I've worked as an emops engineer in a core ziml team in order to streamline mlops Frameworks and furthermore I worked in several us-based companies as a contractual roles as a data scientist and not only this I love teaching my courses has got millions and millions of views throughout the internet and I've helped several thousands of students in order to get their first paycheck or their first job and but why you should consider learning deep learning and what is the core problem which is coming into the deep learning content throughout the internet now it is all of the companies are asking for a deep learning skill set into a particular candidate and to be honest people think deep learning extremely hard and pretty hard to understand and it's my personal opinion I feel that because of the instructors on YouTube it's becoming a little bit hard to understand it's not because it is very complex I agree that the whatever things are little bit difficult or hard to interpret it but if taught in the right way deep learning is the most I think easiest subject as compared to even core machine learning I will teach you deep learning in a very core way from very scratch we will mathematically do every iteration by our mathematics so that you understand okay this is how the flow is going and this is how each steps is helping your model to learn or become better let's get started with our course [Music] hey everyone welcome to this first lecture on linear algebra so today we are going to talk about uh vectors and we'll be exploring vectors A bit okay so uh first we'll start off with what is linear algebra why do we even bother to study this as some of you all already are familiar with algebra you you just want to uh just re refresh your memory of your algebra or you wanted to uh you're from very scratch and then s so that's why let's let's start with the definition of linear algebra so I've written one definition of linear algebra is it's the mathematics of the data yeah you heard me correct I saw this definition online and I found this a very basic definition to tell you uh rather than taking too much of uh mathematical terms is algebra L linear algebra is the mathematics of data and why I'm saying it's mathematics for data because uh L linear algebra contains of matrices and vectors so so these uh these these two are the language of the data so whatever we are going to study you which you will see in your machine learning or deep Learning Journey whatever you are going to study so that's why we we just say that is it's the mathematics of the data because in machine learning data is so much of uh basic component or or a mandatory component the same way the we use algebra or L lar algebra to work with the data mathematically okay so this is a simple definition of linear algebra over here so let's get started with the first uh first uh the component which first thing which we'll study in this video is vectors okay so so we'll start with the vectors so we starting with vectors so if you if if you have any definition of a vectors please please please stop stop this video pause this video and then go down in the comment and please tell me what are vectors so the the we will start with very scratch uh you can assume a vectors as an arrows as an arrows we we have an arrow so these These are the geometric inition of a vectors so the definition of a vector can be it's it's an arrays of a numbers okay so vectors are arrays of a number or a tuple of a numbers okay or or or you can you can take it as an arrays okay so you can consider a vectors vectors can be arrays of a numbers arrays of number or you can consider this vectors uh as an as an arrow you can consider vectors as an arrows as an arrows or you can consider a vector as a tuple of a numbers you can consider ve Vector as a tuple of numbers okay so these are you you can just imagine this an an array is maybe an array can be 1 2 2 3 this is called this is this is called the draw Vector which you'll study you can safely ignore this so this is this is also a vector which is special type of vector which is called the row Vector this can be tle of a numbers or it can be arrows okay so so so the way I like to represent uh vectors or to make you very very much comfortable with it is to to make you familiar with an arrows so this is the geometric division of a two-dimensional Vector so let's see how the vector looks numerically okay uh in in terms of mathematical so let's see let's let's see how the vector look so let's name the vector as U okay so let's let's name the vector as U equals so let's store 2 and four so U is a vector where you have the elements so the the numbers inside the vector you you enclose into a a square bracket so here here is two and here is four okay so why I'm saying we have this is this this is like a an arrow so the first element is called X component and the first element is called X component and uh second element is called a y component it's called a y component so let's let's see how this looks on on a graph paper or or to or an X and Y plane so let me plot let me plot that so 1 2 3 4 5 1 2 3 4 and five okay so this is my y plane and this is my x-axis okay so let's plot this Vector onto this uh X and Y plane so so X component is 2 and Y component is four okay so X component is two and Y component is four so here's the point and it passes through the origin so so this is your vector U okay so this is your vector U where you have 2x 4 where two indicates the X component and four indicates the Y component okay so this makes sense I hope so okay so and vectors are arrays of a numbers or you can say the the Tuple of a numbers which which which consist of numbers where it is it has only it is here here our Vector is two dimensional okay but but uh here you can have M number of a rows here you can have in vectors you can have M number of rows you can have M number of rows but and you can and in vectors you have only one number you have only one column okay so so so you can have any number of you can have M number of rows for for for example for example let let me show you a vector U Can Be A B all the way around to the N okay so it can be n dimensional Vector so here our this Vector is two dimensional Vector this Vector is 2D Vector okay and geometrically I showed you by plotting on this XY plane that this is the the two two dimensional plot means first of all X we we we we we go through x x is the four the two units on the x-axis and the four units on the Y AIS and then we and then we uh taken the from the origin and that point okay so so this is this a graph for you but let's take an example eight so this is our Vector a where you going store n dimensional Vector uh it's not key that you should only have two two dimensional Vector you can have n dimensional Vector okay so so but but showing you geometrically you can show three three dimensional Vector geometrically so you can just draw a straight line over here and this is your Z okay so you can plot a three-dimensional Vector so so so you can plot it for example you taken K as a vector and you can plot 2 32 on this threedimensional plane or the three threedimensional graph but you can but you can kind plot your you cannot plot your four-dimensional or or five dimens Vector over here so for geometrically understanding I have just just showed you how this Vector looks like but it's not a manner that you can only have a two-dimensional vector or or only three dimensional Vector you can have nend dimensional Vector because scientist or researchers most care about your uh or nend dimensional uh your numerically rather than uh most of of course they care about geometrically as well but uh but I just showed you it's not possible for me to draw a FL four dimensional and show you how this how how how we are going to plot but but geometric contribution of a vectors are are we can plot it like this and for example for example you you you have a 2 4 3 so here your K here your K is 3x 1 so first is what are the number of columns which usually denote as three number of columns and you have only one row of course in vectors you can only have one row okay uh okay so so so here you have X component here you have X component here you have y component and here three is your Z component which is in threedimensional and so on okay so this is how you represent vectors so the whole so the whole intuition about vectors so I hope that you understood what I'm trying to convey you over here okay so so so let's so let's see so let's see uh so let's let's go further into understanding uh some more intuitively one last examples of a vectors to to get us what is trying to convey and and and it's it's it's it's much better for for for us to understand okay so I'm going to just just just draw an X and Y plane over here so I'm just just going to draw an X and Y uh X and Y plane like this X and Y and I'm going to take one I'm going to take 2 3 4 5 and six 1 2 3 4 5 okay so this is our X and Y plane now now what what I'm going to do is make it for for example you can you want to plot the vector one 2 okay so how so here you go one unit or X's unit or one unit on the X because this this is your X component this is your X component this is your X component so you go on X unit over here so we'll we'll go till here okay and then two units above so this is your this is your final vector v okay is a final vector v and and to for denoting the Y always the name of the vector should be in lower case with one Arrow Above So this this indicates that it's it's a vector okay so this is this is how you can you have to practice so just try to plot a vector U where X component to be 4x4 okay and a vector can be nend dimensional it can be 6 7 8 9 it can be in dimensional so this is this is how you this is what the vectors are okay so vectors are a two polar array of numbers which we which we just shown shown you today okay so now let's see how we can take out the length of a vector so for example you just draw this Vector so the v v Vector so how do you how can you take out the length of this Vector it's it's a good good good good good way to think about this okay so what I'm going to do now what I'm going to do now is to just remove this and show you uh so take another another example so I'm I'm talking about how do you take out so let's take one example that you have a vector U you have a vector U you have Vector U we have a vector U where your X component is four and 4 by 4 and four okay so here you have here you have a two-dimensional Vector two-dimensional Vector so you can you can also write w with the member of R2 okay so uh so this is the this is the w u is the member of two-dimensional real numbers okay so this is this is this this is your example so what what you want to do is plot the or just let's plot the vector on this okay so 4X 4 I should see over here so X unit four over here and four units of Earth okay and then let's touch this point I think it's wrong bit but no problem this is so how how are you going to take out the length of this Vector this is a good good question to ask to you so for taking out the length of this Vector okay so what you can do here you can see here you can see here you can see that this is also four units this this is also four units and this is also four units so I'm just just going to change the this this is also four units this is also four units okay and you can see this forms a right triangle right triangle at a 90° okay so this forms a right triangle so you know this this is so you know this so which is four units you know this which is four units and you know the this x which is which is here the base four units so here height is 4 units and your base is four units can't you take out the hypotenuse okay or the vector length by using Pythagorean theorem of course you can take out so you can use the Pythagorean theorem you can use the pytha Pythagorean theorem Pythagorean theorem to take out the hypotenuse so for taking the hypoten so the the u² equal to b² + h² okay so you don't know U Square you know b square which is 4 you know the H Square which is 4 okay so U ^2 = 16 + 16 which = 32 and then U ² = 32 now want to U equals to square < TK of 32 that is actually 5.6 5 and and and nearest 100 okay so this is your length of the vector this is the length 5.65 is the length of the vector so so the norm you you usually say the norm okay so in in linear algebra terms so the norm of the vector which is equivalent to length of vector is equals to 5.65 which is your an which is your the length of the vector okay so I hope that you understood what I'm trying to convey with geometric intuition over here okay so what if if we have n dimensional Vector okay so what if we have n dimensional Vector so we will we will see what if we have n dimensional Vector but but let's see let's let's go further let's understand a bit more intuition about um the the how many number of elements what what are the terminologies and then we'll see how do we take out the length of an N dimensional Vector okay so uh just as a notation or terminology or to to remember so that everything is clear everything is clear uh the elements in the vectors are the dimension of the vectors I'm not saying that uh elements it's the the number of elements in the vector is equivalent to the dimensions of your vector okay so the number so the number of elements elements here is numbers okay so elements are usually the numbers like four is an element four is an element okay the dimension are the dimension dimension of the vector of the vector okay so this is the first terminology so here here you can see that you have Vector U which has a and b so here it has two two elements so this so so here is two elements so here it the U the vector U is a 2d Vector is so I I am forgetting is a 2d Vector is a 2d Vector two dimensional Vector so you can plot it if it is three threedimensional you you'll be having a bit difficulty in plotting in a threedimensional plane but you can plot it if it is four-dimensional you cannot plot it on on over here okay so this is a number of elements are the the are the dimensions of your vector okay the next terminology vectors can be n dimensional as I stated the vectors can be vectors can be can be n dimensional Vector n Dimension and dimensional so you can have the U Vector as a b or all the way around to the N okay so here it can be n dimensional Vector so as I stated that it should not be only 2D Vector it can have a n dimensional Vector okay so so as as I left you hey hey how you're going to take out so you take out the length of this Vector U by just uh by just using the Pythagorean theorem but how you how you are going to take out the length of vector U which is an N dimensional so how do you take out the length so the norm of U so I'm talking about this U the norm of U is a square root of U1 2 + U2 2 + u3 squar all the way around to the U N squ okay so it's it's just equals that not nothing much deeper which which we have talked or i u i s okay so this is the this this this is what I want to convey over here this is what I'm going to convey okay so you can actually actually think think about it and in in that way that you want to take for taking the length you just Square U square plus u Square all the way around to the whatever the number of elements in your okay so over here what we were doing we can simply do like this we can simply do over here if if you want to take out you can simply use this of course you are it's just related to some Pythagorean theorem okay so over here you can just add a square root of your b² + h² which is b square is U1 H square is U2 okay means this one four and four okay and then you take out so it's just equivalent whatever whatever we had seen over here okay so so the same way we take out the the dimens or the length of our Vector uh U okay which is an N dimensional Vector I hope that understood till now whatever I taught Okay so so so so let's so let's see so let's see uh bit more so we have studied the how what are the vectors how do we take the length how do we represent the vector v vectors can be n dimensional so now let's see some of the let's do some of the operations on our Vector okay so so we'll start doing the operations on our Vector so I'm just just going to give a headline uh doing operations doing operations on Vector okay on Vector so it's a it's a very great it's it just not too much hard it's very very easy so so the first operation which you want to do is addition of a vector so how are how are we going to do the addition of a vector the first component is addition so you are given you are given so let me State the problem you given the vector the vector a which is 2x 2 and you're given the vector B which is 4X 4 got it you want to what you want to do you want to calculate you want to calculate you want to calculate Vector C by adding a + b means you want to calculate the vector C by taking out the the the uh adding by by summing this to Vector okay so how are we going to sum some this 2x two and 4x 4 means so what you can do you can do the element wise you can do the element so so you can what you can do c c will will be equals to 2 + 4 and 2 + 4 2 + 4 2 + 4 and then also 2 + 4 okay so two element wise addition so it will be nothing but equals to 6 by 6 okay where you add the one you when when you add the vector 2 by 2 by 1 + 2 by 1 2 * 1 which is the the dimension of the resulting Vector will be also 2x 1 okay so what what do you do you you 2x 1 uh which is the dimension of your a vector and then 2 2x 2 * one where the there is two elements and the one one one column so that is also B and then the resulting Vector is 2 * 1 okay so it is the 6X 6 so you get you calculate C to be 66 where your X component is 6 and Y component is 6 so the 66 is your resulting Vector so your resulting Vector C is a two-dimensional Vector okay some of the things which I want to highlight is your dimensions of your both the vector which is a and b should match okay if for example for example if your a vector a vector a vector is 2x two and your V vector is 446 then you try to add it the resulting Vector will be undefined will be undefined okay so your dimensions of the addition of vector should match okay otherwise it is undefined operation okay so so this this is this is what I to convey over here so let's see how you add the vectors how you add the vectors uh how you add the vectors geometrically so we seen the numerically how we add the vectors so let's see how you how you add the vectors geometrically okay so let's let's do something let's do something is uh we will do now some geometric because people tend to understand more geometrically rather than numerically okay so we'll understand geometrically so here I'm drawing I'm going to draw One X and Y plane like this okay X and Y plane okay so so let's see so let's see that you that your vector a that this is this this this is this is your vector a so let's let's let's keep uh let's make it total okay so this is so this this this is your vector a and this is your vector B this is your vector v okay this this is your vector v and this this is your vector a you want to add this two Vector geometrically speaking you want to add add this Vector a plus Vector B okay you want to add this so when you do this geometrically okay so what what you going to do you're going to take this Vector a so I'm just just just going to say you take this Vector a take this Vector a and put the tail of this Vector the tail of this Vector onto the onto the top of the vector B okay you put the tail of this Vector onto the top of the vector B okay and then you put this this this one onto the head of this okay what what you want to do is uh take this VOR whatever whatever the length okay whatever may be the length you just put it like this I'm I'm not drawing correct but no problem in that so the length of the length of this should should be same as whatever you are doing you're just taking this a vector and putting it over here okay you just um you're just taking this Vector a and putting this tail onto the head and and then you are and then you are putting it over here and then what what you do you take this vector v you take this Vector B and put the ta onto the top of a and put the and and just match this uh with this okay so take a vector B and then you just put it like this I think it's not correct too much but this is how you are going to do you first to take the vector a put it on the the the tail of that on the head of B and then you whatever the length you just uh you just put put that over here and then you take this and that and then put put put this state onto the head of this and then you match the heads okay and then and then and then the head the the the resulting Vector so the resulting Vector so the resulting Vector I'm just just just going to take this so so you just take you just this is this will be this will be this will be your resulting Vector we are blue one in the case A+ b a vector plus b Vector okay so this is this this this is how you add the vector geometrically and then it makes sense as well okay if Ison don't worry let's see one more example to make more clarity okay so over here which you let's plot so this is how you just take this and then you draw and then you are done with this and then you attach the the here's the tail on the origin okay and then you attach and then you draw a straight line on like that okay so this this is called the parallelogram method okay so this is called the parallelogram this this is called the parallelogram method for showing the geometrically for showing the geometrically let's see triangle method that that will make make more even sense okay so that will make more even sense so here is your vector a here is your vector a and here is your vector v here is your vector v okay here is your vector v you want to add this up so what do you do you simply you simply what you what you don't do anything extra you simply attach this to make a triangle to make a triangle like this and then this resulting Vector is your addition of the two vectors okay so this the resulting Vector which is which I'm going to highlight with this is your resulting vector and this is this is the this this this is how you do the addition geometrically speaking okay so this this is called the triangle method this is called the triangle method for addition of two vectors okay try to play with it a much much more better way so that it could make sense to you as well okay so try try try to play with it try to draw some diagrams of it and then show and then try try try to little bit juggle with it and then then and then you will better better understanding rather than uh just seeing okay so you can you will be getting some assignments on this as well so you can approach the assignments uh in geometrically speaking okay so so let's see one more operations which is the vector subtraction Vector subtraction okay so I'm just just going to show you Vector subtraction subtraction so what this Vector sub subtraction will do so let's say you want to subtract Vector U minus minus vctor V okay so that will be simply we can frame it as an addition of a vector U plus minus B okay so then it will be much easier to show it geometrically speaking okay got it so what I'm going to say over here is you have a vector so you can simply do like this for showing it geometrically now it will very easy to show it geometrically like this by adding of the vector okay so what what what what you do you just make this and then just for for showing it geometrically speaking okay so for example you have a vector U which is 2x3 and we have a vector and and you have vector v which is 1x 1 subtracted so resulting Vector the resulting Vector will be 2 - 1 and 3 - 1 that will be nothing but equals to 2 okay it's 1 2 okay so X component y component so this is your resulting Vector okay so let's let's see how it looks like uh so it it would make more sense to you as well okay so let's assume 1 2 3 okay 1 1 okay so let's plot the U Vector so I'm just just going to plot the U Vector which is 23 okay okay so okay it's two now so it's 2 three so okay I think I done wrong it's 2 three so just going to okay so this is your vector U so this is your vector U this this this is your vector U and you want to plot the vector v v so this is your vector v this this is your vector v okay this is your vector v and the resulting Vector is 1 by 2 okay so this is your resulting Vector which is your after subtracting okay so after you subtract uh uh this from this the vector v from U okay so that will be your resulting Vector that makes sense geometrically speaking as well okay so this is how you do the vector subtraction geometrically if you don't understand geometrically it's no worries but it's not more than tough which which which is is very very easy not more than tough okay so this is your vector addition and Vector subtraction so let's see the last concept which you which I to make you familiar with is is is Vector scalar multiplication Vector scalar multiplication Vector scalar multiplication so I'm just but but first of all what is in scalar what is an scalar so scalers are give scalar such as constant or a numbers for for for example four is in scalar two is in scalar one is in scaler or or or or anything okay so this is scalers are just constant okay so it is it's a number but in algebra or linear algebra term we call it as a scalar okay so it plays an important role when we stud about L linear combinations or linear Transformations it plays a very important role so so so this is so what if if you multiply a scalar for example you have you want to multiply scalar a times the vector U you times the vector U so let's let's take the you you take the scalar a to be two and the Vector U as a 2x2 okay so multiply 2 * 2x 2 so it 2x2 so it do the element wise Product 2 * 2 4 by 4 that will be equals to 2 * 1 okay so Dimensions will be 2 * 1 that that is simply the vector scalar multiplication so what it actually does in geometrically speaking it stretches the vector so it doubles the vector so for example you have this Vector 2x two okay and then you multiply with two then it will be doubled then then it will be stretched okay then then it will be stretched means transformed the vector okay or stretched the vector by by following the linear structure it stretched the vector like like this so this this this was your initial Vector so after applying the a times the vector U that is stretch which is your final Vector after applying your uh the scalar and Vector multiplication okay so this is your stretch stretched Vector so please please please please ensure that it's stretched so before applying it was U and after applying is doubled okay so this is this is this is the vector scalar multiplication and the reason why we are studying these because this helps to build a very good foundation the stretching the geometrical speaking this helps a very good foundation when when we talk about Transformations con combination combinations uh igen values I vectors these these plays an important role in that so so this was uh so so so we have seen a bit about vector and scalar multiplication the last concept the last two concept which I want to introduce to you is the unit vectors is the unit vectors is the unit vectors and the zero Vector okay so the unit Vector so unit Vector is any Vector with a length one so so the definition States the unit Vector the unit Vector is any Vector is any Vector for for example any Vector okay any Vector with length one whose length is whose length is one whose length is one that's a unit Vector okay what is a Zero's Vector Z's Vector is whose length is zero whose length is whose length is whose length is zero whose length is zero okay so that is the zero Vector you usually denote the z z Vector in very bold way okay that is the zero vector and that is the unit vector vector and vector okay so these are the two basic basic very very basic ter terminology which you need to know about okay so so we have seen a lot about our vectors in this video so I hope that you understood every Everything whatever we had have have have a talk on this so just just just to make sure that everyone understood this so what what what you actually do so let me show you one one more example of that so so for example you have this okay so you have this Vector a and you have this vector and you have this Vector B okay so what do you do for addition of these two vectors you can just you can what you can do you can just uh make this make this a triangle okay but let's approach within using a parallelogram method okay so what do you do you put you take this you take this vector and put on the this tail out of the head and then you just draw a parallel to this and you take take this and then you take this and then you join the and this this A+ B is a result in Vector okay so this is how you go further into approaching these stuffs and I hope and maybe you can solve it using a triangle method to show you how it works geometrically speaking okay so I hope that you understood whatever I'm trying to convey you over here okay so I hope that you understood geometric intuition the triangle method what the unit what are zeros what are scalers what are what are what are vectors and etc etc etc okay so I think that we are done with this lecture um on it's 30 minutes so I hope that you understood vectors what are vectors uh so you you can find the notes of this the lecture notes maybe all of these in the description round box below uh the the or in LMS and the assignments will will be also related to this will will be released at the end of this week and I hope and I really really really hope that you understood this and if if not please feel free to ask the question in description down box below or in our Discord server we are we will be very really happy to answer your questions the next announcement is you can simply uh the next the next lecture will will be based on matrices okay so we'll talk about a bit about matrices okay and then and then we'll talk about after after completing a bit of mat matrices we'll talk about linear combinations then trans linear Transformations okay so we will be studying these things so don't worry we'll gohe very slow pie and very easy way okay so thanks for seeing this video I'll be catching up you in the next video till then bye-bye and have a good [Music] day hey everyone in this lecture we'll be talking about mates in our pre previous lecture we talked about vectors and I really really really hope that tat to understood about vectors I know this these are very very easy concept for you but but uh let me tell you these sets a foundation when we study about combinations or Transformations and other other stuffs so that's how we are prely focusing on this so from this video we'll be starting stepping up a bit difficult note from matrices and then we'll talking about some some matrices operations and then we'll talk about some properties of matrices multiplication and then I will just end up with this video with Matrix Vector products and then I will show you the wide results at the end of the video that the linear combination of the column Vector okay so we'll be talking about I'll just introducing a notion of a linear combination so that in the next video is totally based upon your linear combination so that's why I will just give you a taste of linear combination at the end of the video so let's get started with matrices so today we'll be talking about matrices so let's let's recall a bit about vectors so so vectors are uh n dimensional where where it can have n and where where it can have n number of rows but only one column okay so we were having this it can have one two all the way around to the n and here this is n * 1 so the shape of this vector v the shape of this vector v is n * 1 and it can have n number of a rows and one number of a column and this is a vector specifically we can call this as a column Vector okay so so so we specifically call this as a column Vector okay so so it is given a new name and when we when I will introduce you a notion of a matrices then then we will use extensively in the in the later videos but but this is also called the column vector and and for example your vector can be in this 1 2 3 all the way around to the end okay so this is called the row Vector this is called the row Vector okay okay so this is called the row Vector so this for this these are the two two things which I want to in introduce to you so we'll be covering this again just after we complete the matrices okay so let's start with what are matrices so matrices are a are a set of numbers or or a multi-dimensional l okay where it can have n number of rows or n number of rows and M number of columns it can have a multiple columns and multiple rows okay so in vectors we we in vectors in vectors we were have having only we were having only n n rows and one column which is the example of the example will be 1 2 3 okay so this this this is an example of a vector v okay so the matrices so the matrices can have can have n number of rows as usual but n number of columns but can have M number for column so as an example we can make that Matrix a equals to 1 2 three so one column 2 3 4 second column 3 4 6 third column so it can have M number of M number of a row M number of columns and and and N number of a rows and N number of a rows okay so here the shape of this a is 3x 3 uh where three is number of a rows number of a rows and this one is number of a columns number of columns okay so so over here this this first first indate the 3X3 Matrix so this is the example is 3x3 Matrix so the formal definition of a matrix is is matrices are a set of numbers okay mates are a set of numbers arranged in a rows and a columns which is n rows and N columns so to form a rectangular array okay so here here it on the rectangular array in other words matrices can have n number of columns and M number of rows okay so let let me write a formal definition of matrices over here so and the definition which is just let me write the a good definition of this so that everyone can can Define what a matrix is so matri so here's the definition of a matrices where it it it is arranged in a rows and a column so I'm going to give the name of a rows going to give the name of a rows to n and columns to M so to form a rectangular l so for for for example we can have here it can have a b c d e f g h i okay so it's can have any here we have this is this this is an example of 3x3 Matrix where we already have one two three rows and one two three columns okay or we can say in other words it's going to have n number of rows and N number of columns M number of columns okay so I'm again I'm saying n is for number of rows number of rows and M is for number of columns number of columns all those are in small small letters okay so this is your form formal definition of a matris and the lecture notes is in description box below please go there and assess your lecture notes for you to better to to just revise in the meantime Okay so so so just let's write a formal notation of how the maor matrices are so that so that it it is uh easily inter interpretable so I'm going to make a matrix a going to make a matrix a where I'm going to make a matrix a where it I'm going to make this a11 so the how do we assess the the first element so here it is in first row and First Column so this a is in first row and First Column so that's why I written I and Z okay so a i and Z indicates I what is the row number and g z what is the column number so for example for assessing the elements so over here you have this you have this Matrix we have this Matrix a and what you going to do is assess three so how do you want to assess so the you the formal the formal assessing things is a i j or or yeah so a i j where I indicates the row number and J indicates the the column numb okay so over here you can see the three is on is on we start with 1 2 3 okay not from zero so 2 2 A 2 means a A2 means the row number is two and the column number is also two column number is two that is nothing but equals to three okay so here's how you assist so first of all you write the row number then you write the column number so I indicates the row number and J indicates the column number okay so I hope that you understood what I'm trying to say so it is telling go and this is the first element where is first row and First Column then it is second first row and second column then uh all the way down to the first row and N column okay or M column okay and over here it can have 8 2 1 okay so second row First Column second row second column all the way down to the A2 m m second row M column okay uh so uh this can be a31 a32 a3m here all the way around to the a N1 so n is number of rows or a N2 all the way down to the all the way down to the a n m okay so here it is the formal notation or or a definition which we can write over here which is the formal notation for writing uh so so this is a matrix okay so here's how I developed this so it can have M number M number of a column A rows n number of rows so 1 2 3 4 all the way down to the N which is n number of rows and it can have only it can have only M number of columns okay it can have only M number of columns so this is your formal formal formal definition which you have which I have given to you for uh matrices okay so I hope that you understood now what I'm going to talk about is I introduced a notion of a row vector or a column Vector is it so I introduce you so can you just go and just type me what is a row vector and what is a column Vector so let's take one example so let's take one example is you have a you have a matrix but just just just one thing that the vectors are subset of matrices okay so the the the vectors are a n * 1 Matrix or n * 1 matrices okay so the vectors are a subset of matrices okay so if you if you extract this extract this extract this row that this is just a vector okay so what I'm going to do now is uh is going to just make a vector make a matrix a make a matrix a which contains just just don't relate I'm just taking examples I'm just taking examples you have a vector a I'm going to store 1 2 3 4 5 6 okay so this is my this is my 2x3 Matrix okay so so what you do you take the first row okay and then stored in another so C okay that is 1 2 3 okay so what is C over here C is called the the the the the the the Matrix with one row okay the Matrix with one row is called the row Vector this is called the row Vector this is called called the row Vector The Matrix which has only one row is called row vector and the Matrix with only one colum which is nothing but called a column Vector okay so so so if we if you take this 1x4 okay in D that is 1x 4 it has only one column that is nothing but equals to column Vector which is nothing but is column Vector okay I I hope that you are understanding whatever I'm trying trying to tell here so so these are the size of a matrix over of size of a matrix where we have row vector and what is row Vector row Vector are nothing but the Matrix with one row with one row is called the row vector and the matrices with one column is called the column Vector okay so if you take one example so just just just just assume that this is your this is so this is your first of all row Vector sorry column column Vector because you have uh you have so so you have sorry column vector so this this this this one is a column Vector which is V1 so V V1 over here is a column Vector okay so if we take this one if we take this one so here you have only one row so that is B2 which is your row Vector okay so this is this is what the notation of the notion of a row row vector and column Vector means and I really really hope that you understood about this okay so if not please please feel free to ask ask a question below where you're stu please please please use Discord server or whatever that's the doubt support which is provided to you so that you can get most out of out of this course and if you need any guidance for absolutely free please feel free to reach out to me via email Discord comment we can get on a meet to help you solve the doubts it's it's for Okay cool so let's so we have we have seen what some matrices are so just just want to Recaps youate everything so matrices are a set of numbers arranged in a number of rows and number of column s to form a rectangular array where it can have an m n number for rows and M for mango number for columns okay so so the notation for for the definition of geometrically over here is the a so I I have made this as an example the show showcase shoe and we have we have seen some of the Matrix size where you where the the the terminology which is where the Matrix has only one row that's the row vector and with the Matrix has only one column that is a column Vector okay so I hope that you understood what I'm trying to convey over here cool so now let's talk about so now let's talk about so now let's talk about some of the operations because in pre previous video we talked about vectors and then operations so the same day I'm going to talk about operations on a matrices okay so operations on matrices just just just just just going to show you show it to you operations operations I think my handwriting is not good too much no problem operations operations uh on matrices okay so you want to do the operations on matrices so the first operation which you're going to do is Matrix and a scalar multiplication okay so what I'm going to do is Matrix Matrix scalar multiplication okay not a not a big deal it's very very easy to understand okay so assume that you have a matrix a you have a matrix a you have a matrix a 10 6 4 3 which is your nothing but a 2x2 matrix okay 2 rows and two columns okay and you have a scalar you have a scalar a which is nothing but two okay so you're given these two now what you want to do is uh you want to multiply uh a a okay so we multiply a scaler with a matrix which nothing but 2 * 10 6 43 okay that is so what will be the result please anyone please please please feel free to to pause this video so what you're going to do is multiplying a scaler with a 2x2 matrix so what will be the result anyone please please please in the comment okay how how do you tell okay no problem but please please free to ask say in the comment box I will be very happy to see the if you're still here so the answer of this is first of all what do you do you you simply do the element wise product with this scalar okay so 2 * 10 which is nothing but 20 which is nothing but 20 2 * 6 which is nothing but 12 2 * 4 what it is 9 no it's 8 okay 2 * 3 which is nothing but six okay so the resulting the resulting Matrix is 2 * 2 Matrix so what you do you transform or or or not a transform I would say you just multiply a scalar with a a matrix and and then you get a 2x2 matrix which is the same size of your a okay so 2 * 2 = to 2 * 2 which result when you multiply with any scaler okay so what it does is simply do the element wise produ product okay so the so the formul notation for this is you you multiply C with with a okay and a have I and J row okay so what it will do it will simply M do the element wise a c * a i j that is nothing but what it will do first of all it will uh so in more more notational terms it will just for for for example you have a vector you have a a scaled a and you multiply with c d e f okay so what do you do you simply uh a * C A * d a * E and A * F okay so this this will be the form this is this this will thing and then you'll be getting some values 2 by two okay which with your values okay the same this is this is what it is doing so this is your formal definition formal definition of your Matrix Vector multi multiplication sorry uh scalar matrix multiplication and I really really hope that you understood this let's go on to the next operation which want to see let's go on the next operation which want to see is addition of our matrices okay so the next operation which we are going to cover is addition addition of matrices okay additions of a matrices so let's Zoom so we are you are you are given a matrix a you give a matrix a which is 1 3 1 uh 1 0 0 okay and you have and you are given Matrix B okay and Matrix are always written in a capital letters make sure and the vectors are always in small letters okay with one over here uh and the scalers are also in like this yeah so that's B is 05 75 okay so what you want to do you want to add these two matrices A + B add this two matrices so this is the operations which you want to do this is the operation which you want to perform so how how are you going to perform this operation how are you going to perform this operation so for performing this operation so for performing this operation you will just what what you will do you wanted to just have this 13 1 1 0 0 plus 05 750 what you what you will do you will nothing but uh 1 + 0 at element y some okay in scaler you doing so you will do the same 1 + 0 3 + 0 1 + 5 okay element wise sum okay 1 + 7 0 + 5 0 + 0 okay so this is this this is what you do and then you'll be getting your answer which is 1 3 6 6 and then you getting 8 5 0 that is a resulting U Matrix okay which is also 3x3 Matrix okay so what you do you simply do do this and you're getting a 3X3 Matrix okay so so when you what you do you you just added a 3X3 Matrix 3x3 Matrix and then and then you and the resulting Matrix is also 3x3 okay so so this is how you do the addition of a matrix and the formal definition for this I which which which which I can state so because definition is very very important for fundamentals for for making your fundamental strong okay so the definition is your given a matrix a which can have a uh b c d e f okay that is nothing and your B is also some some kind of k g i h o p okay so add these two so you what what you will do a + k b + G C + I D+ D + h e + o e + O and F + P that will be nothing but equals to 3x3 Matrix okay so I have just taken example your your Matrix can have n dimensional the can have any number of rows and any number of columns but there are some constraints which you you need to there are some properties like Dimension property which you have to take care while adding the matrices okay so it can have it can have uh ABC you can it can have a 10 x 10 matrix the size of the Matrix and then you are adding the 10 x 10 matrix with another 10 by 10 matrix and that would resulting in another 10 by 10 metri okay so that is the that is the thing uh so some of the property which I want to highlight which you all have to focus on so some of the properties of a very very very very uh simp simp simple property that commutative property commutative property so addition of a matrix is commutative commutative and all of these is written in your notes please please please feel free to write see from there if you if you wanted to just just just revise it up okay uh in in future but I would highly highly recommend to complete this video V plus so you so you have a matx say you have a matrix you can do B+ a okay uh the next thing is associated property your your your your addition of a matrices are associ associative as well okay so associative associative property associated property I'm not writing a lot of properties over here but the one who are important I'm writing over here A + B+ C which is nothing but equals to A + B plus C and all of these can be proved very very easy the proof is very very easy not a hard please feel free to search on internet about the proof it is very very easy associative commutative it's it's the proof are available on internet okay so so the last thing what I'm which the first property is commutative the second is associative the last one is dimension property the dimensions are be the dimensions the dimensions the dimensions of the dimensions of your of your uh uh the two two two Matrix would be same The Matrix the matrices would be same the matri should be same okay the dimensions okay if it is not then then it will be undefined your operations will be undefined Okay cool so so now we have we have we have talked about one one of the operation which is addition of a matrix and I really really hope that you understood addition you you understood a scaler but one thing which I'm going to spend some some two minutes talking about is you may be thinking yeah you do we really really need to know about these stuffs uh so I would say yeah you need to know about although you don't need to just worry about how I'm going to code it you can actually develop it from very very scratch not a big deal but there are some libraries like numai which would be in this course we'll be using in this course or or pytorch that'll be doing using uh the LI library for S computation for addition of a matrix for multiplication of a matrix which they handle which they are very efficient okay because in real world your Matrix are not 3x3 Matrix they are they are they are billions size size is millions okay Millions by millions so so they are they are very very large so so so Maj mag multiplication with your own for Loops are very very time taken okay so that's that's that that's a big deal okay so your your time complexity will increase as your input size increases okay so that's a big deal so you we are learning this to understand the inner workings of our of our function so that we can we can know how how our algorithm is doing and how everything is working behind so that it it becomes very easy to debug something or you get some error or or or to to have very good or decent knowledge of what your code is performing Okay cool so addition of a matrix is also done now let's you can do you can do the same with sub subtraction of a matrices please try it out by your own the next thing which which I'm going to spend some time talking about is matrix multiplication okay bit bit I'll spend some time talking on this okay so the first the next next thing which I'm going to to talk about is Matrix uh Matrix Matrix multiplication Matrix Matrix multiplication okay so this is your uh next operation which is which is one of the most important important uh important what do you say uh the operations which you which you need to learn okay so you given a matrix a you're given a matrix a I'm going to take very easy example 1 7 2 4 okay and you're given a matrix B given a matrix B which is 3 5 3 2 okay now you need to now what you need to do you want to multiply Matrix a matrix B okay so how do you do you may be thinking here you here is 2x two here is 2x two is 2 by 2 1 * 3 7 * 3 2 * 5 that's that's that's that's that's not how you do okay so the matrix multiplication the way you do is like this you take the first row of that a matrix you take the first row I'm going to change my pen you take the first row of that a matrix and multiply with the multiply with the First Column of the b b Matrix multiply with the First Column with the B Matrix okay so so here's here's how you do so resulting resulting M Matrix c will 1 * 3 okay plus taking the dot product of your of your row Vector times the column Vector so what you what you are actually doing is taking out the dot product dot product which you'll see in our later videos don't don't worry about that dot product of this this is your because if if you see this is your it this it it has only one row okay so of a vector of a vector of a of a vector of a row Vector of a row row vector and column Vector column Vector so this this is your column Vector where it has only one column so specifically you're taking out a DOT product of a row vector and a column Vector what is dot product so dot product is element one so what what you do you simply multiply and add it up but element wise adding and add it up okay so over here what what what you want to do you one * 3 and 7 * 5 1 * 3 7 * 5 and then you will add it okay now take this again this row again this row with this row row vector and multiply with this column Vector multiply with this column Vector so that that will be nothing but equals to 1 * 3 + 7 * 2 okay now what do you do now what do you do you you now you have this The Taking of the dot product of row row vector and the second column VOR which is V2 okay now what do you do you go go go further into this the second row Vector which is 2x4 and then you do the same 2 * 3 dot product of the 2 4 with row Vector with the column Vector okay 4 * 5 okay and then 2 * 3 4 * 5 2 * 3 2 * 2 okay so 2 * 3 + 4 * 2 okay that will be nothing but equals to C which is nothing equals to uh 1 * 3 which which will be how much 1 1 * 3 which will be 3 + 5 35s okay 1 * 3 3 + 14 2 * 3 6 + uh 20 which is 26 2 * 3 6 + 8 which is nothing but 14 okay cool so this is your resulting and then what you do you simply make your 38 17 26 14 will be your resulting Vector which is 2x2 matrix which is the 2x2 matrix okay so when you multiply with this you will be getting uh your favorite the after after multiplication of the Matrix this this is your answer of this of your particular question got it and I really really hope that you are that you understanding what I'm what what whatever I'm trying to say but you it may you you can use you it can have any dimensional but some properties are there okay so let's let's visit the property I'm just going to constraint that Dimension property is very important in this so what dimensions should match to be so that the matrix multiplication is not undefined okay so so the properties of for MRI multiplication so the first property which I'm going to talk about property the first property is for for example you're given a matrix a b and c so these are three Matrix which are n byn Matrix which are n by n Matrix where you have n n byn Matrix okay so where you have n rows and N columns okay so the first thing which holds is commutative property of multiplication of this multiplication does does not hold okay so M multiplication is not commutative is not commutative is not is not okay so when you multiply a b ba a which is which will be totally wrong it can be proved it can prove it it can prove very easily then the next property associative property of matrix multiplication is is is there okay so associative property associative associative property is there okay so a b + C which is nothing but uh a okay so I think it's I have written for this distributive I written for a c which is nothing but equals to A B C okay it can prove rigorously and of course you're multiplying it over here okay it can it can prove very very easily it is distributive property distributive property this this it is also distributive property so what you want to do a B+ C that that will be A+ a c and you can prove these you can prove this very very easily which which can be found on internet the next thing is the most important Dimension property Dimension property Dimension property so the dimension property is you can have M number of a rows you can have M number of rows you can have any number of rows but an N number of columns okay this is for the dimension of a matrix a your dimension of a matrix B should be your your your number of a rows should be same as uh number of columns in that Matrix okay times K any okay so your resulting will be M * K okay so so it makes sense as well if you have you can have M number of rows at the a a matrix but you can and N number of columns okay but with here you can have only n number of columns n number of a n number of rows sorry n number of rows okay so here you have two 2 by two so here it has two and the resulting will be the resulting will be 2 by 2 Vector sorry Matrix okay so that the the the resulting size will be this okay so this is your dimension property of your matrix multiplication cool so we have talked a lot about Matrix and and and and you seeing that you're are going and then and you are seeing that that you are going bit up bit bit little bit little bit up so the last thing which I will end this video which I promised you is is talking about Matrix Vector product is a matrix Vector multiplication okay so what I'm going to talk about is Matrix Vector multiplication yeah so let's let's let's let's do that then so let's do that so you have a matrix a you have a matrix a which is nothing but which is nothing but so I'm just just going to define rigorous very very definition of Matrix Vector product so I'm just just just going to write it very very fast a11 a12 all the way around to the A1 n and a12 a22 A2 n uh a M1 which is is N1 okay N2 okay that be A and M okay so this is your Matrix this this is your Matrix so it is having M number of columns and N number of a rows okay this is your Matrix a and you want to multiply with multiply with uh a vector a column a row L row a column Vector okay X2 XM XM okay so to do the multiplication of it so how do we do it how you how you how you will will will you do it so you want to multiply a matrix which is a which is M * n Matrix times the vector a scalar uh sorry Vector col color Vector X which will be the definition will be so so what what it be the answer of this so how will you perform the so for performing the operations is nothing but equals to a11 X1 so what you are actually doing you can take this a11 okay and you multiply this with this multiply this with this okay so what you actually doing what you what you are actually doing you're are multiplying you're multiplying the the column Vector the column sorry row Vector sorry row Vector to the column Vector okay element wise and adding it up okay so the dot product between your row vector and a column vector and adding it up okay so a11 plus you're adding plus over here see a12 X2 plus A1 3 X3 all the way around to the A1 M A1 M * x m XM okay then you do the same A1 2 X1 + a uh plus A2 all the way around to the A2 uh n okay so you're multiplying with X whatever the X so what what you're actually doing you're multiplying you're taking this column Vector taking this column Vector so row vector and multiplying with this so you are taking on the dot product between taking all the dot product taking all the dot Dr taking out the dot product and giving your answer that's it okay taking taking out the dot product of between the row vector and the column Vector so let's see with one of one of the example so it it it would make more sense so here you have - 3 0 3 2 1 7 - 1 9 okay and you have a vector column Vector 2 - 3 4 1 okay so what what will be the output so the answer will be uh minus 3 * so so we taking this you're taking this and multiplying with this okay - 3 * 2 + 0 * 3 okay - 3 it's minus + 3 * 4 + 2 * 1 it's yeah it's 1 okay now you go for the 1 * 2 1 * 2 + 7 * - 3 + - 1 * * 4 + 9 * 1 that will be nothing but equals to after after you add it everything after calculating okay after calculating that will be a and b which is 2x 2 2x 2 sorry 2x one vector so which is a column Vector which is 2x one so after after after doing the Matrix Vector you transform you what what you do you have this R4 R4 we is four dimensional vector you transform it to D after doing the after using this V using this Matrix say you transform it into a a and b which is the 2x1 okay which is from R4 to R2 okay so here you transformed using this Matrix a so so so we'll see in our four videos that matrix multiplication are a linear transformation okay so we'll see in our later videos but but now as of now I I hope that you understood the mutrix Vector product okay so in the next video which what I'll be showing you a wired thing over here or not a vired thing a very useful thing over here is the linear combination using the help of Matrix Vector product so I will take an example of matric vector product as a linear combination of uh of so that it would make more sense and then we'll complete the linear combination now in in our next video and then and then we'll end up uh this uh uh so we'll be completing and then then we talking about the linear transformation I really really hope that that you understood this I'll be catching up you in the next video till then bye-bye have a great day and and please please and one more thing attendance is you have to mark your attendance so please please feel you to do so bye-bye have have a great day [Music] okay so welcome to this lecture in this lecture we we'll be discussing about linear combination of a vector so and this is this is one of the most important concept as you will go further and you will understand okay so it sets up a very strong fundamentals to mathematically understand or to see the Deep learning or machine learning into a linear algebra point of view so this is one of the most important concept which we'll focus on so in this video we'll talk about that specifically so uh as so I'll in this video I'll just give you I'll be giving you some definition of linear combination and then I will giving you some some examples and then we're talking about a matrix Vector product as a l linear combination because in previous video at last we talked about Matrix Vector product so that's why we are in this video we'll be talking about the as an example taking that as an example for linear combination of our Matrix Vector problem okay so so the definition States so the definition of a linear combination States uh so let's let's do something let's start with an example let's start with an example and example States and an example States key that you have so first example that I want take is you have a scalar you have a scalar and you multiply with the sum Vector U okay and then you have and then plus two with some vector v okay and that will be some resulting Vector that will be some resulting vector which is just the scale version of u and v which is nothing but uh for example D okay so that is the resulting Vector of our uh after applying off first of all we we added it up sorry mult M multiplied and added the vector okay so that that will be the resulting Vector so the resulting Vector is a linear combination of vector U and a v okay so again listen me that what you we we have taken this example and and this in this example we have one scalar three and then we multiply with the vector U so for example we can have a vector U to to be 2x two so when when you multiply three times okay that would be nothing but 3 six six that would be the six okay 6 six okay plus you have some some Vector B it is nothing but 4 4 okay so 2 * 4 so when when you multiply or do the scalar scalar Vector multiplication that will be simply element wise product so 2 * 4 which is 8 and 2 * 4 which is 8 okay so when you uh now you add 8 by 8 okay so that will be 8 by 8 that will be nothing but 8 by 8 okay so the resulting Vector will be 14 by 14 okay so that the the resulting Vector will be 14 by 14 which is your D which is your vector D so this the vector D is a linear combination of your vector U of your vector U and of your vector v okay so the the the resulting Vector is a linear combination of these two Vector which is u and v so I hope that you that that you are understanding what what whatever I'm trying to tell and this 14 is just the the 6 + 8 which is okay so this is the D is a result resulting Vector uh after after you do so what you specifically done is multiply there is some scalar so there is some scaler so you are given any number of a vector so the I'm writing the definition so what you specifically done you're given you're given any number any number of any number of vectors we given any number of vectors and the linear combination linear combination linear combination of the vector of the vectors so you are given any number of a vector but the linear combination of this Vector means these vectors are simply the result of are simply the result when we multiply when we when we multiply when we multiply each Vector each vector by a scalar scalar and add the vector vectors so so how you get the linear combination is you are given any number of vectors okay so you are given any numbers of vectors and the linear combination of those vectors the given vectors is simply the result is this in this case the result of when the the result when you multiply when you multiply each Vector which is you are you are given u and v over here so when you multiply each each vector by a given scalar which in in this case is three and two okay and add them up that the resulting Vector is your linear combination of those vectors okay the formal notation which I can write is you are given a vector a you are given you are you are given a vector B you are given a vector C you you given a vector D okay so you want to find out the linear combination of these vectors so so the linear combination of these vectors will be simply uh when you multiply with some scalar okay with some scalar so I'm just going to write it uh uh a okay so this so let me write a different name of this so I will just so your given given vectors given vectors are are I think uh v u g okay so this these these three are your uh given Vector okay uh now what you do uh you simply multiply with some scalar a okay so so what what will be the so we want to ask what will be the linear combination linear combination linear combination of these vectors so what is linear combinations so the definition states that a linear combination is the result when you multiply the given vectors by some scalar and add the vector okay so that is the resulting vector so what you do simply multiply a scalar with a vector plus b with the U Vector plus C with a z Vector the resulting Vector D which will be nothing but your linear combination of v u and G okay so that will be the resulting that will be the linear combination so let's see more of the example to get comfortable with this so that you could get uh you could get a good feeling okay this is the linear combination Okay so so another example can be you can have a vector U you can have a vector U and you can have a vector v you can have a vector v so so what you do you it can be like this any number minus one * U Vector which is a scaler plus z b the answer whatever the resulting Vector will be will be the linear combination of these two vectors which are the given vectors okay so it can be fraction as well your it for the the the scalars can be fraction and fraction as well 51 7 by 11 it can multiply with some some some some some Vector U and 1 195 by 2 with some vector v the resulting Vector the resulting Vector the resulting Vector will be your linear combination of these two vectors which is U and which is V okay so whatever the resulting Vector is will be the linear combination of u and v Vector okay so let's take one formal example so that we we could understand this much much better okay so one formal example of linear combination say you given a u uh say you given a vector U say you given a vector u u which is which is uh which is two two dimensional Vector which is a 2x1 vector or we can say it's a column Vector minus 5 by 0 because column Vector is what what the the vector the the matjes is with only one column and over here this is we have only one column so so so so that's why it's called a column Vector so that's why we are telling it to the column Vector please see the previous video to help us understand to help you understand much more better way okay uh and you have a vector and you have a vector v which is 02 okay so you so you given a vector u and v okay this is also a column Vector this this is also a column Vector so what you want to do is to take out the linear combination of these two vector and and and the linear combination can be uh we can multiply this this this is this uh U Vector so what we can do we can multiply any scalar with this U Vector plus any scalar with this V Vector we will'll be getting the linear combination of these two vectors so let's take an example so let's let's take one example you multiply 1 with - 5 0 okay - 5 0 plus uh you multiply you you you have a scalar one which is place of v and you have 0 by 02 okay that is your VV and the resulting Vector which will be nothing but - 5x2 okay and this is the linear combination of these two vectors and you can you can you can go ahead you can try it at different different scalers and and the same will be so you triy first you can try different different scalar with two with the a - 5 0 + 2 it it can be any number 02 that will be nothing but when you do do this to which is - 10 and then - 10 0 + 0 4 that will be nothing what equals to - 10 4 so this is - 104 is a linear combination of these two Vector yeah you heard me correct this is the lar combination of these two Vector as well as this is the lar combination of these two vectors you heard me correct it can be you can multiply with some scalar 4 okay with -50 0 Plus plus I think okay you can do with one uh 02 the resulting Vector whatever the resulting Vector after doing this a will be the linear combination of these two vectors you heard me correct yeah exactly so your linear combination will be is it it it can be it it can be anything okay after the whatever means you take any scaler multiply with a given Vector uh you will and add it up you will be getting a linear combination so linear combination cannot be over uh some some finite over here okay except some exceptions which are there okay so over here linear combinations are said to be a vector if there exist a scalar a b okay and then whatever the resulting Vector will be will be the linear combination of those two vectors okay do not think that linear combinations can be one one is line combination of any two Vector can be only one no it can be anything it it can be any number of a linear combination of that two Vector what you need to do simply multiply the vector with some scalar A or B whatever and add it up the resulting Vector will be your linear combination of those two vectors so again I'm writing one formal formal definition so we have already written one formal definition but let's write one definition which will give you a more idea about what we have seen so far so the so the definition States so the definition States a vector R okay so a vector r a vector a vector r a vector R is said to be the linear combination is said to be a linear combination a linear combination a linear combination a linear combination of a b C okay uh a a vector is said to be a linear combination of a vector a b and c Etc okay so a a vector R is said to be the linear combination of these given vectors A B C which in this case this was u and v in this case this this was u and v these are the given these are the vectors okay these are the given vectors so the same way a b c d these are the the given vectors okay if there exists if if there exists if there exists scalers if there exist scalers x y z Etc whatever the means whatever how whatever the number of your vector is such that such that your resulting Vector is equals to x a plus YB zc all the way around to the n so so a vector R is said to be the linear combination of these vectors of these vectors if you multiply the scalar with a given vectors respectively respectively and the then then the r is said to be the linear combination of these two vectors of these all the vectors so again I'm repeating what's the linear combination means it simply means that it simply means that the the vector R is is is is we can call it as a linear combination okay how we can call we we are given a vector we are given these vectors we are given these vectors and if there exists exists some scalar and such that in such a way that your that the resulting Vector is equals to the multi the the the r is the is the multi when you multiply a scalar with a vector and add add the add the vector the resulting Vector will be your linear combination of that uh given vectors okay so let's take one example one simple simple example is uh is let's you want to you want to say is 1x 4 so you're you're you're given a vector U so I'm just going to take take one example so that everything is um make you understandable so your your your example you you're given a vector U you're given a vector U which is - 5 which is a two two dimensional vector and you given a vector v you're given a vector v you're G given a vector v which is 02 which is 02 so you want to show you want to show is is your 1x4 which is your R the vector R is the is also a linear combination is also the linear combination combination of vector U and vector v so you want to show is this Vector is the linear combination of these two Vector okay you want to show this you want to show this this is a problem so you want to show this is this the resulting which is r 1 by4 is the linear combination of these two u and v Vector is you want to show it so how you going to show it so for showing it we we will see the systems of equations solving the systems of equations later on but we can I will just give you a tool so so 1X 4 so this will be the resulting Vector so if there exists some scalar which which is 1x 5 so this is this this is my scalar times your vector - 5 0 plus there exists the second scalar times the 02 so your resulting Vector is 1x4 and hence and hence this 1x4 is the linear combination of this U and of this V Vector okay so I hope that you that that you are able to understand what's I'm talking about the linear combination of these vectors I hope so okay for a linear combination of a vector U and a vector v and I have given you also the formal definition of a linear combination so so let's let's see one terminology the terminology States terminology States terminology states that terminology states that the constants or the scalers which is here 1.5 this is two so these These are called the weights of the given Vector so in we we don't call it as a scalar we we call it as a weights rather than calling the scaler so if you have seen your if if you have seen your uh your hypothesis function so you have a Theta so that's so so so that is so that is the weights weights of your vector so the same way over here we don't call it as a scalar we call it as a weight of that u and v Vector okay so I hope so that you are getting a point uh to towards linear algebra and viewing machine learning or deep learning into the view of linear algebra okay so I hope that you are understanding so let's take one example to understand the weights so for example for example for example 1 by - 5 is your linear combination of vector 1 4+ 1 one when we multiply with the some weight this is called the weight so three that will be the this is the linear combination of these two vectors these two vectors okay and over here over here that 1 4 1 14 which is a vector and vector and 1 1 with weights with weights which is - 2 and 3 okay so 1 by - 5 is the linear combination of a vector 1 4 and 1 one with weights Min - 2 and 3 we don't don't we we don't call it as a scaler we even call it as a weights of that thing okay so I hope so that you are understanding whatever I'm trying to say you over here so I'm just going to talk about one last thing if I have a time I do have a 10 minutes time so what I what what I can talk about today is is the next concept which is the span of a vector okay so I'm just just just going to just make you familiar this is this this is just very easy so the we be talking about a span of a Vector I'll just giving you a small introduction to span and in the next video if possible I can show you some some some geometric intuition of a span okay so what is a span so let me let me write it span of okay so what is span span is a set of all the possible linear combination of a given group of vectors so for example we showed you over here we showed you you can have a multiple linear combination of a given Vector u and v you can have a multiple L your combination are you getting me so you can you can you can have multiple linear L linear combination of that given vectory U and B or u and v so the same so the group of all the or the set of all the linear combination of a given group of Vector in this case u and v is the span of those vectors I hope so that you're getting me okay so let me show you within help of example or or before that I'm going to give you a formal formal definition of this span because I think I I I TR some definitions also so SP definitions gives you a clear way of thinking this so I'm going to just give you a definition this this definition of a span the definition of a span the definition of a span is the set of all the possible the set of the set of all the possible all the possible possible linear combinations I'm just just going to write linear combinations linear combinations L linear combinations of given of given group of vectors okay so group of vectors what do I mean with this that uh how the the like in in this for example you have a vector U and you have a vector v so you want to take out the linear combination so these are the group of vector to for for which you want to take out the L linear combination okay so that given group of vectors is called the span is called the span of those vectors the span of those Vector which is a group of vectors span of those vectors so we'll see see one example to help us understand is much better way so let's take one example of that the example which I'm going to take is this is not a very hard example just just going to take a simple example but given a vector U 22 and then you have and you given a vector v which is 1 1 okay so this is the two vectors which is and any any and you can take out the linear combination you can take out the linear combination where when you have 2 3 4 9 10 may you another Vector is also the same stage same size okay so but should be the same Dimension so you can take out the linear combination of this just you need to multiply with a scaler and then add it up and then you'll be getting your L linear combinations so so over here so over here so over here if you so first of all let's take out some set of linear combinations so what I can do if we can multiply this U Vector with some weight I'm not talking I'm not taking the name of scalar just for practice a good practice so we can multiply with some weight two uh 2 two which is and here I can multiply with I think one one one and you can add it up that that will be nothing but 44 going add it up I'm just being transparent so that everyone follows the same is because I think trans being transparent uh just for a sake of it's it's very very important so that the conceptual CCT Clarity uh will be very very easy for you all okay so this is the the first the first for this is the let's take an example this is your l l linear combination so uh your L linear combination is this for for example we written V okay no not V we have already already given so this is your first V your combination let's take out the second or third let's take out a second you can multiply 3 2 2 plus I'm just sticking anything 2 1 1 which will be nothing but 3 6 6 + 2 2 that nothing but 8 8 okay so this is your H which is a second L linear combination let's go on taking the third so this which is the last one for us 4 2 2 3 1 1 which is 8 8 + 3 3 which is nothing but 11 11 okay so this is your I which is your another Vector okay not I let's let's let's give it as okay not J as well let's give it as a key okay let's give it as K Vector okay so these three are just you can take out any you can you can take out lot more l combination you can just go ahead just multiplying with some scalar and then multiplying with some vector and then adding it up you'll be getting your linear combination of that two vectors so I'm not I'm not arguing you with that so I've just taken three linear combination so these three the five 58 5 five this is your G just taking z z h and k is the span it's a span a span of vector U and B instead of saying lot of Lear combination so you can just say okay that this these are the span of those two VOR so span if if you see geometrically speaking if you see over the it just your your V combination span hold to the space okay so please see some for some visualizations to to from three three blue one brown they give but I have given a geometric intuition that all the possible linear combination is called the is called the span of those vectors so the notation for writing this is just a span of the vectors V1 V2 V3 is written as a span of V1 B2 V3 okay just I've shown you over okay please see the notes the notes are also given to you just for your own good okay so this is this this is what I talked about a bit about span and I hope that you really really understood this let's go on a last topic of this video is Matrix Vector product okay so Matrix Vector product so what I'm going to do is uh I have already talked about Matrix Vector product but just the last thing which I'm going to just talk about so I do have time so let's talk about Matrix Vector product Matrix Vector product so so let's say you're given a vector a you're given a vector a where you have some a b c d e f g h i okay this is your uh Matrix a and then you have a scale uh then you have a vector X then you have Vector X which is X uh for example one X1 X2 and X3 okay so you want to take out the majri vector product so how do you take out so it will be simply what you do so what you do you will simply uh you will simply this is your a x which is simply nothing but uh you you take take this you take this column you take this column you take you take this column which is the column vector and you take this uh sorry row Vector you take this row vector and you take this column Vector you just multiply it or you can say you can take we'll talk about the dot product just don't worry so what we can say we take out the dot product of a row Vector of a row vector and a column vector and a colum Vector okay so what you are what you are specifically doing is taking out the dot product between between two with between a row vector and a column Vector so what does dot product mean means it is just element wise product and sum it all up okay so that will be nothing but uh that that will be a * X1 + B * X2 + C * X3 okay and then when when you add it so let me write a resulting Vector as well okay so that will be a d okay so after you multiply and add it so that's is just a DOT product of of of of two vectors of two of of a row vector and a column Vector which is this we'll talk about the we'll talk in detail about the dot product and a transpose later on okay so and then you do the same D * X1 + e * X2 + f * X3 I'm going to do the same G * X3 + H * X2 plus okay this one uh I * X3 okay so you'll be getting e f so whatever the answer is and this is your final uh Matrix Vector product product okay so which you have seen in in our previous video but I'm going to relate as a linear combination so I'm just just going to do what what what I'm going to do just see over here so you have a vector a you have a vector a you have Vector a where I'm going to what I'm going to do a b c d e f g h i j k l okay and I'm I'm going to multiply this this Matrix with a vector with a vector X1 X2 X3 and X4 okay so we have this a vector and X sorry aain Matrix a and a vector X okay so what you going to do you just just going to do the same thing you to multiply ax okay so I'm going to show you how you can do how how I'm going to represent this okay so what I'm going to do is to categorize this Matrix in into column Vector different set of different set of column Vector okay so we can take this we can take this the first First Column the First Column and say this okay okay this is the so you take it a V1 okay take the second column you say this V2 take the third column you say this V3 take the fourth column you say this V4 okay so so so specifically you you you are not taking as a as a as a as as a as a DOT product or or the element wise product of row and column Vector here what you will do here what you do categorize your Matrix a into a different uh into into a set of uh column Vector which is V1 vs2 V3 and V4 okay and then when you multiply when you multiply ax ax which is nothing but what you will do what you will do you will simply do uh X1 * V1 X1 * V1 okay X1 * V1 you multiply all X1 with this a okay X1 * V1 okay and this is your scalar this is of course your scalar so let me write in small X1 * V1 okay so V1 V1 + X2 * vs2 + X3 * V3 V3 + X4 X4 X4 * V4 okay so that whatever will so this is whatever the result will be for example R will be your linear combination so just listen what I'm try what I'm trying to say you converted this now what you do you take the you take this X1 and multiply X1 * X1 * X1 * C and X2 * X2 * X2 * X3 * X3 * X3 * and X4 time so what you do you given a weight of these column vectors so the resulting Vector will be the linear combination of the column Vector a so this this R will be the linear linear combination linear combination of vector a okay of a vector uh of of of a m of a column Vector a okay of a column Vector a so so so these this result resulting Vector will be the linear combination of the column Vector a of the column Vector I just just just have read column Vector column Vector which is V V1 V2 V3 the V4 so you have seen me how I done this as is to to Showcase you in in the form of linear combination so I hope that you understood a lot from this video and I really really hope that you will uh try to try to do the uh try to mark your tendance as well because it is an LMS so try to mark it your notes are in the description on box below please feel free to assess the notes uh it is very very important to to to work from that uh uh and and also and in the next video we'll be talking about uh the the linear transformation where we'll be introducing the notion of a transformation when when we multiply a two Matrix so that the the that that is th a transforming one Matrix into from one dimensional space to another another dimensional space that is a transformation which we'll talking about in the next video and then and then I hope so that we'll be able to complete the uh chapter number one in some days and then I hope uh it is it is much clear to you as well uh the next the one of the announcement that I want to give is please please please share the video and and mark your attendance in your in your quizzes okay uh uh by by going to the assignment Tab and your LMS if if you are in LMS so please go there and please please please try to try to search for some some some problem set or try to search for some resources although you don't need I talked a lot about more than enough okay for for your journey so so thanks for seeing this video I I I hope that you enjoyed this and you have taken your own notes please feel free to assess the notes on the description I'll be catching up your next video till then bye-bye and have a great [Music] day okay everyone welcome to this lecture on L your transformation so in this video will be is specifically talking about linear transformation in the pre previous video we talked about Matrix Vector product or or or in or a linear combination which which was the very fundamental concept and we also talked about span of a vectors now in this video we'll be talking about linear transformation one the most amazing concept or the beauty of algebra or linear algebra which you will ever see and also this this sets a very Foundation of your of your algebra skills or after study as of now okay and and it is used extensively in the field of deep learning uh when when you read research papers or or when when you staring some algorithm so if you want to understand by the point of view of linear algebra then then I think uh linear transformation is one of the best thing uh to study and and this is a compulsory topic to study as the most most most concept is related to this but you may think hey you just transformation so can you just Define what a transformation is I just want to you to search on Google what a transformation means and then put that in a comment box please give the time stamp so that I could I could know okay you you're here okay so please go on YouTube try to search about transformation and then come back okay so transformation like like just transform something or or do something to that function or or if you know about function so linear transformation can be thought of as a functions okay as the new name given when when we deal with in lanar algebra okay so lanar transformation is nothing or can be just thought of as a functions can be thought of thought of thought of as a functions can be thought of as a functions why I'm telling this in in functions what you do you give some input value you give some in input value and you want your F and you want your F to map this input value to the output value y okay so this is this is this is what the L transformation is doing linear transformation take some some some some some Vector as an input maybe two two dimensional Vector which is two dimensional Vector it wants a function it wants a function T that maps from n dimensional to M dimensional Vector okay this is the definition of linear transformation again let's will see see some of the definitions to help us more clear what actually linear transformation means but in but but in but in big picture uh linear transformation can be thought of as a functions like it takes some some some some sort of vectors or vectors and transforms from one di to n dimensional Vector to another dimensional vector or a different space in that uh plane okay so this is the this is the basic thing which you need to study about so u l transformation can be thought of as a function that takes some some some some vectors or or just transforms that just transforms this Vector from n dimensional space to M dimensional space and that can only be possible with a t which is a function T so the functions are the one which we take some value and Maps input and input values to Output values but in transformation we take we transforms xn means n dimensional Vector to M dimensional Vector okay so that's a linear transformation with you the definition so let's let's let's start with an example so that we could understand it's much much much much more better way okay so just I'm going to write a definition and what in what we do in a case of functions and what we do in a case of uh linear transformation so in functions we take some values we take some values we take some values input values and we map our input values and we map our input values input values to Output values okay so to make a function f that take some Val Val and output the square of X which is Parabola if if you plot it out okay then that in in case of linear transformation we what we do we make a function T we make a function T we make a function T we make a function T that takes the transforms the transforms from from one vector space transform the vector the transform the vector uh from from RN from RN and this is the RN from one from n dimensional into a vector into a vector to RM okay so it just transforms one one one vector from one one vector space to another Vector space okay so this is the basic thing which you need to which which will which will see I just prove prove you out this equation so why it why it seems to be legit so let's start with with an example so that I could just prove you that what whatever I'm telling is correct okay so let's let's take one example let's take one example say you have a you you if for example if you multiply example if we example will be say you multiply your M by n Matrix so you have an M by n Matrix you have your M by n Matrix this is your favorite Matrix you have your m m byn Matrix and what you want to do is simply multiply with a column Vector which is n * 1 which is a column Vector which is n * 1 so we'll just multiply this this Matrix with a column Vector n * 1 so let's let's multiply it out with the column the column Vector n * 1 okay so here it has only one column and the resulting Vector what we are specifically doing we we are taking we are multiplying our Matrix with a vector okay and then that that will be nothing but your n by one column Vector M by m * 1 column Vector okay m m * 1 column Vector which is your which is resulting Vector resulting Vector that that will the resulting column Vector resulting column Vector okay so so what so what it does so what it does it take it it it it took your it took your um just is to took or or or in other words we see over here that an N M * n Matrix that an M and M * n Matrix transforms an n * 1 Vector into an n M * 1 resulting Vector which is another space okay so using this Matrix we transform this Vector into different Vector space like this okay so for example let's see some some example to make sense here we are taking the taking the the the the the Matrix Vector product which we have already seen in our previous videos so let's consider consider this Matrix a let's consider this Matrix a as we as a 3X3 Matrix so 1 2 0 and 2 1 0 which is nothing but 3x three Matrix okay now I want to show you want to show show you want to show that by matrix multiplication by Matrix Matrix Matrix Vector multiplication or that matrix multiplication matrix by matrix multiplication by matrix multiplication you want to show by matrix multiplication a transforms this this Matrix a transforms transforms transforms Vector n r R3 vector and R3 mean is which is the which is the column Vector which is which is three threedimensional Vector like this have Vector which is a three dimensional x y z which is from R3 from from in from R3 to R2 or to R2 okay into into a into into a vector into Vector in R2 R2 so you just you using a you want to transform this Vector X X you can transform the vector X and then that will into into an R2 which is nothing but your X and Y one two two dimensional rather than being a three dimensional that's that's when we call linear transformation of a matrix okay so let's see so with with an example you have a you have a vector so so R3 so over here R3 R3 are a vector of a size is a vector of a size 3x1 which is a which is the 3x1 threedimensional vector while Vector R2 R2 which you want to transform you you want to show is 2x 3 2 * 3 which is the two sorry uh 2 * it's it's 2 * 1 it's two dimensional Vector so it's trans it's using the a you want to transform your your your your vector means you using a or or a transform you just show that a transforms the vector means from R3 in R3 into a vector R2 okay which you want to show up okay so when you do this when you do this so this this this is of size 2x1 which is which is your uh which is your row Vector I I think it's column Vector column vector and then you if if you multiply a which is your 2x3 Matrix which is a true two okay it's 2x3 okay it's 2x3 matrix by a 3x1 vector by a 3x1 vector by a 3x1 vector the resulting Vector will nothing but 2x one so you just showed using the what you do using a you multiply this a with uh which you want to show means R3 and then the resulting Vector which you can see that you showed okay using a you transforms the vector in R3 which is uh three which is this one into a vector R2 which is 2 2 okay or or or a which is a two-dimensional Vector okay so that that is what it is telling so let's see one of one of one of one of the example just to numerically show you so so 1 2 0 2 1 0 and and what you and and you have this metrix a and you want uh um and you and and and you have a threedimensional vector XYZ and you and this is an A and this is a vector X and you want to show you that a transforms of in a trans means uh what do you see the a transforms and and a matrix which is from R3 into an R2 okay using a you want to transform R from R3 R3 into R2 so that will be nothing but x + 2 y and zero of course we don't write it out and and of and and 2x + y okay and that and then it it will be after after you add it up add it up it will be either A and B which is your R2 okay so that is the following that what you done you simply transform your f one one one vector space to another Vector space using a function a or or using a which is your Matrix okay so let's let's let's define it out so what you done what you done you made a function T the transform F and and from from n dimensional to M dimensional means a function T transform the vector M transform the vector RN into a function a function T which transform from RN to RM which another dimensional space got it the linear transformation should satisfy your two constraints the two constraints are the first constraint the transformation t x + y it would be it would be is equals to the TX plus Ty y transformation of X Plus trans transformation of Y and then your trans transformation and then this this is scalar a and this is a vector X and then it's should be and then then it should be a uh a and transformation of X so these are the two conditions which you need to satisfy okay uh these are the two basic conditions which you will ever see okay but but the more form formula which I which I could State over here is linear transformation is the is the the function that transforms your Vector from onedimensional space from RM to RN okay that is your linear transformation of your vectors or Matrix or of of of of of a vector okay so um one one theorem is there one theorem is there the theorem States uh your let T be our let let T be a function or transformation that transform from one vectors RN to RM um which is the transformation of your transformation of X which transforms RN to RM okay so I'm just just just going to write the theorem which I'm not going to prove rigorously but you can prove it you can prove it t equals to R to make a function T you want to make a function T to transform RM to R RM okay uh be a transformation defined by the trans transformation is defined by T of X okay trans you you give uh you give one vector which is of n dimensional and just what it will do using a which is a matrix just transform that Vector means the Matrix Vector product so I just showed you that a matrix Vector product is a linear transformation okay so what you use if even you multiply to u a vector with the Matrix that that that will just give you the linear transformation just it should satisfy the conditions which are listed over here the geometric understanding is also so not it's is very very easy you can you can consider watching three blue one one Brown videos for this I hope that will make more sense there okay so that was a short video On LAN transformation about fth 15 minutes and I really really hope that that you like this video uh in the next video we'll be talking about transpose of AMS and uh and and and and and and and uh uh dot product which which which which is one of the most important concept so let's get on to the next [Music] video okay everyone so today we'll be talking about transpose and a DOT product of a matrix or a vector so we we we'll be talking about that because in the video of linear combination I have taken one example of Matrix Vector product and I showed you how you can represent that Matrix Vector product as a DOT product between the column vector and a row Vector so we'll be talk talking about what does it exactly means uh what is a DOT product and a transpose of a matrix so these are again two most important concept which you will ever see in your journey of linear algebra again it's just one of the basic concept which is very very easy to understand but still it's is very very uh good to know about these things which gives you an extra tools to work efficiently uh and and your deep deep learning problem or machine learning problem or or or other stuffs okay so let's get started so so the what is a transpose of a matrix so I'm going to start with a transpose of a matrix what is a transpose of a matrix so the transpose of a matrix is a kind of operator which flips over the diagonal which flips over the diagonal okay or make all the rows or make all the rows uh for example if I could show you the the the visualization so for example you have uh you want to take you have a column vector or row Vector like this 1 2 so it's if the diagonal is this one the diagonal is this one so it simply flips it okay so it simply flips it then it would be if you take all the transpose of this so it will be 1 2 so what it does it flips over the diagonal for example let's let's take one one more let's take one more let's take one more you have 1 2 3 4 which is a 2X two Matrix now if you take all the transpose of this Matrix so what you do you have this diagonal you have this diagonal you have this diagonal so if you flip over this diagonal if you flip this over the diagonal if you flip this over the diagonal it would be nothing but one if you flip this over a diagonal if you slip flip it there 1 3 and 2 4 okay and two four so what you done you simply flip it and then above going down and down going above so that is what it is trying to tell or or in other words what you can interpret is you make every row or sorry every column as a row and you make every column as a row okay so that you can that you can expect Ed so for example you have another so so if you have another Matrix let's let's take an example that is 6 4 32 and you apply the transpose onto this Matrix so what will be the output so the output will be what you do you take this uh you take this um or you say the row sorry the column or the column vector and make it a row Vector like this make it a row of row Vector six and three and you take this another you make this four and two or other wasse what you can interpret is you flip over the diagonal you flip over the diagonal you make this three above and you make this four down okay so what you specifically doing is to flipping over the diagonal like this okay so you will get the same result as you are doing so it's just not a big deal to understand these things is very very easy to understand okay so I hope that that that you are able to understand what the transpose of a matrix is so let's try a formal definition let's write a formal definition of your transpose of a matrix so the transpose of a matrix the transpose the transpose transpose of a matrix transpose of a matrix is an is a type of operator or operator is a Operator Operator which flips which flips over the diagonal over the diagonal over the diagonal okay so this is the transpose of a matrix so what is does is it the definition states that it is it just flip over the diagonal or to the main diagonal to obtain the a transpose so for example you are so for example you are given a vector a so what you do what you do you reflect a you reflect a you reflect a over its main diagonal to obtain the a transpose or it is just a tech technical verse to flip or reflect but specifically what is doing making the column Vector as a row vector or and and a row Vector as a column Vector that's it okay okay your visce Versa it can be anything so making a colum Vector to a row Vector that's it okay so that is what it is telling and a row Vector to a column Vector whatever whatever seems good to you so a transpose just exactly doing the that okay so for example for for example you want to take you have your Matrix a okay you want to take out a transpose you want to take out a transpose where it either presents your row and J represents your column so now if you after this after applying the transpose at this Matrix now your will be Aji now your column represents your rows and uh rows represents your column again again this is this is very easy to interpret so you have this 2 two 2 two and you have this a matrix what you what what you do you apply the transpose on this Matrix after applying the transpose of a matrix here you are having a transpose okay and you have I which is two or I and G okay you have I and J which when you do that when when you apply the operator this will be nothing but or I could say 0 0 Let's Take This 0 0 okay so when you when you sorry we should not take this 0 0 let's take it as a let's take take it as a one let's take it as a six okay just for understanding okay so what if if you have transpose over here after applying the transpose of a matrix over here your col your your row becomes your column and column and column becomes your row or yeah so so your row becomes your column so your row is the first row becomes your column so after flipping over the diagonal so when when apply so is two two and then your second second column becomes your second row okay one and six okay so what do you specifically done you flipped over or or reflected the main diagonal or flipped the over the diagonal and then you obtain your transpose of a matrix so that is what it is telling so you can do in high dimensional spaces as well for for example you have a m s we have 64 4 9 10 11 12 13 14 15 okay so if you want to apply the transpose on this Matrix that will be nothing you take this take this and put it over there 649 and then you take this 10 11 12 and you take this 13 14 I think 15 okay so this is this this is what you are doing is flipping over the main diagonal flipping over the main diagonal okay to to to or reflect it or reflect it over the main diagonal to obtain your transpose of a matrix so we have we have done a lots of examples so we'll see one or two more and more examples to make you more sense so let's make one or more example so I just hope that you are able to make sense for actually transpose of so that is the transpose of a matrix so it is just what it is doing reflecting a over its main diagonal to obtain the a transpose that set what the transpose doing so if if you take more examples we have a one two okay if you have one two if this is a A or a okay a column uh row Vector if you transpose this a transpose that will be nothing but one two okay uh just flipping over or or or making the row as a column Vector okay the next thing can be for example you have a I want you to solve this I want you to solve this one 2 3 4 apply the transpose on this and see me what the result result would be in the comment box comment box write this answer in the comment box it will very very easy to understand uh so I could say I could see whether you're watching or not okay so that is the that is the transpose of a matrix so just just I'm going going to write out some of the points some of the points please please pause this video and write your answer of a transpose of a matrix so just I I want you to write it out okay so let's see that some some of the properties some of the properties properties of the transpose okay so you want to take out a transpose and then you take out the transpose that so tell me what it will be tell me what it will be tell me it would be nothing but a it would be nothing but equals to A you you first of all transpose it okay and then you take the transpose of that so for example you have a which is 1 by 2 okay you have 1 by two okay so what you do what you do you take the transpose of this which will be 1 2 and then you take the transpose of this which will be 1 2 so these are equals to or not these are equals to so that's why we are telling that is the one of the property of a transpose of a matrix another property of a transpose of a matrix another property of a transpose of a matrix is A + B A transpose which will be nothing but a plus b transpose is nothing but a transpose plus b transpose okay so that will be equals so you have a a 1 2 3 4 okay and you have a b and and you have a b uh 1 2 3 4 you want to add it by taking all now you want to take all the transpose of these three so what you can do you can take the transpose of you can take take all the transpose of this and that will be your answer okay so let's do this we going to take the transpose of this that will be nothing but 1 3 and 2 4 okay plus uh 1 3 and 2 4 okay which will be nothing but 1 + 1 2 3 + 3 6 2 + 2 4 4 + 4 8 okay that will be your final answer which is 2x2 matrix and this is this is what this property States you can even what you can do you can prove it you can prove it extensively you can prove it you have a you take out the transpose and then you and then what you do first of all let's do the same thing you first of all add it up so when you add it up 1 2 3 4 plus 1 2 3 4 okay when you add it up so 1 + 1 1 2 + 2 4 3 + 3 6 4 + 4 8 it is 2 4 6 8 okay that that will be your answer of this particular and then what you do you take out the transpose of this because you added it up now you take when when you when you take out the transpose it would be nothing but 2 6 48 and this is your answer so these both are equal so these both are equal so you can you can you can do ex you can do separately transpose and then add and yeah or yeah or you can or or you can do just first of all add and then take out the transpose both are equ equivalent your answer will be equivalent okay so let's see another uh let's call it another page let's make another page how do we add another page yeah let's add one one one one one one more page so another property is another another property is AB transpose AB transpose which be nothing but equals to B transpose and a transpose okay so you have a let's say let's for the sake of an example let's take a vector 1 2 1 2 and then you take a and you take a v v Vector B Vector you take a b Vector which is 22 okay and then what you want to do you want to take out the transpose of this okay so let's first of all do this so one so that would be 1 * 2 1 1 * 2 which is which would be nothing but 2 2 * 2 which is 4 okay and then what you do you take out the transpose of this so when you when you take out the transpose of this that that will be 24 that will be 24 which is a column Vector okay so that will be your first understand this is your first now let's let's check for equivalent for this so you have B so which is b 2 2 if you if you take the transpose of this which will be nothing but 2 two okay and then you and and then what do you do you first of all do this 1 2 okay which after taking the transpose of this that that will be 1 2 and then when you multiply it out that will be nothing but 2 4 2 4 which which is both equivalent so so we have proved this we have proved this which we have proved this let's fourth let's go on Fourth property the fourth property is you have scalar C and you have a matrix a when you take the transpose of it that will be nothing but C A transpose okay what you can do you can first of all rather than after multiplying and then taking the transpose you can first take out the transpose and then multiply by the C both will be same okay uh fifth the fifth you can you can actually verify this you can actually verify this is not a big deal okay just the way I'm doing okay so you to take out the determinant of a transpose that will be nothing but determinant of a and if you if if you don't know about this please ignore which we will be studying this extensively in our VAR okay you can ignore this the determinant is is just the area of the parallelogram so you can easily ignore this as of now okay sixth one the sixth one is the last one which I'm talking about a transpose minus one okay uh 12 the^ of-1 a - one transpose so these are equivalent okay so you take the transpose and then and then inverse it first of all you inverse it and then take out a transpose both will be equivalent okay so these are some of the some of the most uh used properties in your transpose and I hope that you that you are able to understand this it's not a big deal to understand okay these are the transpose which we have talked about let's go on to the dot product to understand much better so let's let's let's talk about a bit about dot product so what is dot product dot product what is do what it do what it do it do element element wise product element wise product and sum it all up that's it that's what the dot product is doing is to do element wise product and add a summ that's what the dot product is doing so for an example so for example the dot dot product so the so for example let's let's let's take for the sake of an example you have a vector a where you which is your which is your row Vector so A1 A2 all the way around to the a n all the all the way around to the a n and you have a b Vector B row row Vector which is B1 B2 all the way around to the BN okay so this is your two two vectors now what you need to do if if you want to take out the dot product between these two Vector if you want to take out the dot product between these two vectors so how do you take out so what you you simply add a submiss so you take out the a DOT you write dot b is nothing but equals to I = to 1 all the way around to the n a i b i so what it will do first of all a a A1 * B1 + A2 * B2 + A3 * B3 plus all the way around to the a n * BN and your final output will be one is scaler C which is your dot product between these two row Vector okay so that is the the particular the the algebraic definition so this is the algebraic definition of your dot product and or or or or in other words what you can tell is the dot dot product between two vectors is nothing but a b transpose the dot product the product between A and B transpose that that is your algebraic definition or this is your formal definition of your dot product product so for example for example you have this and you have another Vector like this your output will be 8 a b c d e f your output when when you take out the dot product between these two A+ d a * D + B * e + C * F and your output will be a scalar okay so if you have seen our Matrix Vector product Matrix Vector so we have we were having we were having so let's let's see which we have seen already in our previous videos so let's say let's say let's say of for for the sake of an example you have a b c d e f g h i okay and you have a vector which is X1 X2 and X3 you want to take out the product between these two so what you do you categorize this into a different categorize this into a different what do you say uh the the the column vector or sorry row Vector yeah column Vector V1 V2 V3 then what you do you take out the dot product you take out the dot product between V1 and a vector X so when you take out the dot product between a column Vector with uh with a column Vector so when you take out the dot product between these two that will be your answer which you have already seen in in our previous videos okay so I I don't think that we should uh that we should care about this so I hope that you are able to make sense of these things and please feel free to review the previous video on L linear combination which I talked in detail about these things okay so this is this is what the dot product is simply say simply take off take take out the element wise product and sum it all up okay and that will be your simple scalar okay so that is what the dot product means and if if you have seen our videos on on on hypothesis function which if if you know about hypothesis function of linear regression if we talked about hypothesis function so in hypothesis function what you were doing we were and we were taking out the dot product between our X and W we are taking out the X or a Theta we take Taking of the dot product between X and W that that was resulting in our prediction y hat okay so X was also maybe a matrix or vector so you have a matrix or a vector so a vector of X can be uh uh other met for example 2 4 6 7 and your W can be also 2 4 6 7 whatever these are the weights of this feature and what you do you take out you take sorry it's not matrix it's a vector it's a vector do product so you simply multiply it up you simply multiply it up and then what do you do you simply add it up and that will be one scaler which is your answer for for example 2 * 2 + 4 * 4 + 6 * 6 + 7 * 7 and then after doing plus all those stuff then you'll be getting your answer C which is an answer of this particular question okay I hope that you are able to make sense out of it so we have seen our algebra definition of a DOT product now it's time for seeing the geometric definition to help you more make more sense of your dot product of two vectors okay so let's see let's see of that so let's see uh the do product let's see uh let's take let's take an example you want to take out the dot product take out the dot product between two vectors between two vectors which is Vector a and Vector B which is geometrically speaking I'm going to I'm going to talk about geometrically now I'm going to talk about geometrically so for the sake of an example understand let's take an example that you have a vector like this so sorry it's bad you have like this and you have this okay so this is your vector a this is actually Vector a and this is actually vector v and just one thing if you're a calcul student there is a small fun quiz is it continuous is it continuous function we have a function let's take an example that is a function okay is it a continuous function um if is it differentiable if it is continuous function if is it differentiable that is your question okay so please feel free to put in comment just for just for those who are calc student just tell me just ignore this A and B just say uh just say this this is a function and then just tell me it is a continuous and if if it is then if it is differentiable or not okay this is this is your question to ask answer but the one who is studying Le your algebra please filter stick me with this okay kindly ignore the question which I told you cool so the angle between these two Vector the angle between these two Vector is nothing but the angle between is 59.5 De okay so that is the angle between these two Vector okay so this is the angle between these two Vector so you want to take out the dot product between Vector a and Vector B so taking of the dot product between Ang Vector a and Vector B which which will be nothing but it should be nothing but I would say uh it should be nothing but Norm of a norm of a times the norm of B the norm of B time cosine of theta cosine of theta so Theta the angle between them is Theta so the norm of a so for example let's assume the the length of a is uh 10 okay length of a is 10 and your length of B is 13 so here your vector a is nothing but 68 and when you take out the length is 10 using the pythagore theorem Pythagoras Theorem and then uh the vector v is 5 and 12 when you take out using the Pythagoras Theorem that will be or or the or the norm of a vector when you take the norm of a vector that would be 13 okay so the the the length of these I have already told you okay that is 10 and 13 and when you multiply with a cosine of 59.5 okay so then you'll be getting then what you then what you will get 10 * 13 * 0.575 and then when you multiply that will 65.9 whatever and then it is approximately 66 if you want to uh what do you say take out now if now we got 66 now if you want to take out the algebraic the algebraic definition States 6 8 and 5 12 when you do this 6 + 5 11 8 + 12 20 8 + 12 20 when you add this so sorry uh we are actually taking we are actually taking all the okay that's 6 6 * 5 it should be 6 * 5 why I'm doing 6 6 * - 5 sorry it's 5 yeah 6 - 6 * it's - 6 it's not 6 it's - 6 okay so when you do this 6 - 6 * 5 and + 8 * 12 - 6 * 5 + 8 * 12 and you will get -3 + 96 which will be 66 and this this is equivalent to this and you can also do with this this this one with a high dimensional spaces okay so I hope that that that you are able to make sense out of it and I also hope that uh you you are able to understand a little bit about this okay so that is the dot product between these two Vector just you need to understand the numeric understanding of dot product that element wise product and add it all up that's it okay so I hope that you are able to make sense out of it and I also hope that that you are able to understand everything this was the dot dot product between these two vectors and a transpose of vectors and a matrices and I also hope that that till now you're you're able to understand most of it out of it and I also hope that uh you will utilize this resource and share this resource to everyone that motivates me to work on this content s and so in the next video we'll be talking about some of the types of mates and then and then we'll talk about rank Trace operators determinant I Val igen vectors and solving the systems of equations and then our L linear algebra will be done so I hope that you like this video I'll be catching up your next video till then bye-bye [Music] hey everyone welcome to this next lecture on linear algebra and M 2 and I really really hope that you are enjoying this course first of all I want to thanks thanks you and congratulate you as well that you had first you had completed your first week and I'm I'm very much happy to see so much of enthusiastic students who are watching these lectures without any kind of problems and they're able to understand and leaving their great feedback in the comment box and I'm seeing the watching hours increased so I would like to thank it's giving me a lot more motivation to make these such videos for free as well as um I would just salute you for your consistency and I would also salute for you m utilizing these kind of materials uh who are who are for free and one thing which I wanted to say that we recently in yesterday we uh we we we released our first homework assignment of the previous week which is the the homework assignment consist of the questions from the five lectures previous five lecture lecture 1 to five that is your first week lectures so basically in that we included the homework homework assignments all the questions from the topics which are already taught okay so if you go and see and I and and I would like to thank vayak Vishnu who has contributed 70% of prepar preparation of these questions who is one of the teaching assistants of our uh of our CS M2 so I would like to thank you thank him and I would also like you to thank him on the Discord server or whenever the comment box thanks vayak Bish so it it would motivate him as well he's doing the community work for free so so let's so here is the pro programming assignment to sorry not programming assignments homework assignment where you're getting around 32 questions and these questions are covering from very Basics to a conceptual understanding of the particular subject having in mind to have a good practice of yourself of the topics which are already taught so you may think hey are you you're doing this stuff the reason why I'm I'm making you practice these stuff is the reason one is mainly when you go in deep learning to have a conceptual understanding of what are vectors and how it is performing computations and what are the resulting Vector size and etc etc etc so it will help you to to to to practice a lot and it would also help you to understand the conceptual understanding or on the very depth understanding of these vectors and and and we we also seen some matrices and then we are performing something and then uh I also taught you about linear combinations xx and 3 contains of linear combination where we are talking about various stops over here and the these These are the questions which are very very very very nice questions which are prepared by vayak Vishnu and as well as I had also added some questions over here which are also related to your deep deep learning context and then you finally have a transformation and this this this is also a great amaz amazing uh uh the questions which are there and it will help you in deep deep Learning Journey that how actually the linear transformation works and behind that and then the we we we talked about transposing the dot product of two matrices or vectors okay so that is the specific thing and then we we ask you to verify this property so I would definitely ask you to visit these things it will it Sol it assignment upload it in your LMS learning management system if you have enrolled into that it's absolutely free for everyone please please see the student handbook in the description down box below and go over there and into the LMS and and Summit your homework assignment and then you'll be getting the detailed feedback on what questions youve done wrong and what question you haven't done wrong okay so let's get started with this video so the title of the video you might have already imagined is the types of a mates types of mates so what are the types of matrices that that that we will study and and and some of the types of matrices are maybe not come in your journey of deep deep learning but I would say ke whenever you're studying the something let's study full of that okay so do not let's let's let's not study the partial stuff so let's study full but most of the thing which I'm going to teach is is being used frequently not too much frequently but it's being used sometimes me when when you talk with some great mathematicians or a deep learning Engineers to take these kind of words so it so it should not might confuse you so that's a VA I asked about so the types of matrices so the T first types of matrices which you already know about in other words like row Vector which is something called as row matrices okay or a row Matrix a row so let me add one thing over here row row Matrix row Matrix okay so what is row Matrix so can you define what is what is a row Vector can you just Define it so the row Vector is the is the is the Matrix which have only one row so the same way row Matrix are the one which has only one row so it you you can say just a row Vector uh yeah so the matrices which have only which have only one one row is called the row Matrix so for example for example the example can be 1 2 3 okay so this is the first this is your row Matrix okay so because it has because it it only have only one row so we can mathematically we can we can write that a is equal this is a matrix where a i g where we have a m * n Matrix where M denotes the number of rows and N then denotes the number of columns in this you have m equals to 1 you have m equal to 1 then it is called as a row Matrix so what is a row Matrix row Matrix is a one which has only one row let's let's talk about second second kind of Matrix which is column Matrix column column Matrix so sorry for in in on the eve of diali many the people are just busting up the crackers in India so yeah don't no no problem in that so what is a column Matrix so column Matrix which have only one column so uh and you have you have heard about column Vector so the same way we have column Matrix which only have only one column and these are these are used very frequently in the in the era of deep learning and and and it is very precise to use these names in deep learning to have to to follow the mathematical conventions ra rather than saying it is a it is a it is a call this the shape either shape you can just say that okay it's is a row Matrix or it is a column Matrix or it is a row vector or it is a column Vector okay so here you have a matrix a where you have a matrix a where you where you have M * n which is a size and where the M can be anything you can be any number of rows but you have only one column okay so here here n n should be equals to one so you should remove this and write one to be considered this as a column Matrix okay the next kind of Matrix we should talk about is zero or null Matrix so as you have already uh imagined about this zero all n Matrix and these are very very easy kind of remembering it's not a I'm not teaching teaching any Rockets science is very very easy to understand so so if in all the matrices or all all the elements into that matrices are zero okay so all the all the elements all the elements in the matrices in the matrices are Zer then that Matrix is called zero Matrix okay so for example you have a where you have 0 0 0 where 0 0 0 0 0 0 where you have a 3X3 Matrix and this is called a zero Matrix or sometimes you call it as a 3X3 null Matrix okay so when someone tell you hey can can you can you just tell what types of Matrix this is okay this is a zero Matrix okay when someone ask you hey you're given a null Matrix what what happened when you multiply this new Matrix with another Matrix where the Matrix satisfy all the Mator multiplication property or Dimension property so you you will say okay uh you will say okay what is a null Matrix so null Matrix are nothing but a zero Matrix okay so that is just a zero Matrix which which is given another name which is called null Matrix then the in in that you have all the elements to be zero okay the next is the next is your favorite single ton Matrix okay so in in in Java one of my friend was uh taking a single turn double turn so in the same way we have single turn matrices okay so single single turn matrices so so in single ter matrices your all the matrices are are or or you say you have only one element into that Matrix okay not all the matri you have total of one element in that Matrix okay so that's why we call it as a only one element only one element in the Matrix in The Matrix and so sorry I'm not too much of creative I don't follow the rules of changing the color and then writing it out I should develop that uh thing so let's so let's use different different color then so the fifth one is the fifth one is horizontal Matrix so let's give some examples of this because I I I I I haven't given example of this which is two then it can be one then it can be three these are the for example a is a matrix where we have this is a single turn m matrices Matrix etc etc etc so these are called the single T Matrix you may have St single ton in in your sets or discret mathematics if you have if you have take taken the course on discret mathematics so I would ask you to do not take the course but yeah remember remember the single T either it is not used too much but yeah you you should know the you should know what a single T because when someone ask you you should be able to answer it horizontal Matrix horizontal Matrix that is a that is a good deal so horizontal measor so C and Define if anyone has taken the linear algebra class please feel free to go down in the description box below please so no description it's comment okay so please feel free to go in comment box please write what is a horizontal Matrix and and it's very very easy it's again very very easy so for example so for example uh for example here you have 1 2 3 4 okay and then you have uh and then you have your favorite and then you have uh 6 9 8 2 so I'm going to consider this Matrix as a horizontal horizontal Matrix you may ask why you should consider this as a horizont horizontal Matrix so just tell me the size size of the Matrix is the size size of the Matrix is what uh 2 by 4 two uh rows and four columns and here the The Columns is greater than rows so that's why we that that is a horizontal Matrix so whoever Matrix where the number of a columns is greater than the number of rows then that's called the horizontal Matrix got it so that is the horizontal Matrix where your number of rows Sor not rows number of number of columns is greater than the number of her rows so that that that is one of the example of a horizontal Matrix now let's see some more examples some more uh some more Matrix types which is something called as vertical Matrix so can you tell me what is vertical Matrix just Define me what is a vertical Matrix so let's make a matrix let's make a matrix let's let's let's make a matrix where you have 1 2 3 4 5 6 7 8 9 10 11 12 okay so this is your this is I'm going to consider this as a vertical Matrix now tell me why this is a vertical Matrix the reason why is a vertical Matrix so let's let's count the size so we have a total of 4x3 Matrix and here your number of rows is great greater than number of columns so that's why this is a vertical Matrix in horizontal your number of columns should be greater than uh greater than number of rows to be called as a horizontal Matrix but in vertical Matrix totally opposite of that here your number of rows should be greater than your number of columns okay to be called as to to be said as a vertical Matrix or in formal definition which is Tau in a school is a a matrix is said to be a vertical Matrix if and only if its number of her rows is greater than the number of her columns it's it's it started my school I just I I I just want to thank my school to teach me these definitions no not exactly definitions of vertical Matrix but yeah the definition format or template to tell is that this is said to be this because this okay so I follow the template of my school so thanks to my school H shout out to Sunan cat okay cool the square of Matrix so what is a square Matrix I would say pay attention to this Square Matrix is used extensively whatever we'll study in the like diagonal matrix or or whatever okay like determinant or or igen vectors and IG values we are going to take this Square Matrix so what are square Matrix so let tell me what are square Matrix The Matrix which looks like square is that it yeah so so the Matrix where you're or just wait now here's here's your Matrix here's your Matrix 1 2 2 4 and here the here's your a matrix and here's your B Matrix which is 1 2 3 1 2 3 okay or let let's consider this as a let's consider let's let's cons consider this as a okay so here you have a total of 2 by two Matrix and this is a total of uh what do you say a 3X3 Matrix so here your number of rows matches with the number of columns so the square Matrix are the one where the number of rows matches with the number of columns okay so the square Matrix is which where your number of rows where your number of rows is equals to the number of columns okay so the the square Matrix is is in which your number of rows is equal to the number of columns then it is said to be a square Matrix so again my school form of definition a matrix is said to be a square Matrix if and only if it's it's it's a real number for rows should be exactly equal to the number of a columns in your Matrix okay so that is your Square mat again shout out to Sun uh just just for your information uh this these are the things which are not taught my school till now I'm in class n so till now they haven't taught this but yeah I know the form I I know I know the definition template because I and in in my school I used to study a lot more definitions so I know the definition of this they the definition is starts with this the same way uh this is said to be this because this and that so that's why we got it this so the same way I frame a square Matrix is said to be a square Matrix a matrix is said to be a square Matrix if and only if your number of rows is matched with the number of columns and so thus if it is follows then it's a square Matrix so so we we know about the square Matrix now so let's go further into understanding the the the diagonal matrix so so what is diagonal matrix what is diagonal what what is diagonal matrix can anyone tell me what a diagonal matrix is anyone try it out so all the elements except the principal diagonal or or let's start with let's start I just want to scho that guy who is bursting the crackers I don't want to see him I'm making my video why is bursting his crackers outside oh my gosh no no problem in that as well Okay cool so let's make a matrix 1 2 3 0 0 0 0 0 okay so this is called the diagonal matrix because your your your all the elements in your diagonal matrix except the prpal diagonal this is this principal diagonal and this is the principal diagonal so all the elements into that Matrix except the principal diagonal of a square Matrix so diagonal matrix should be a square Matrix because it is a 3X3 Matrix so it is a square Matrix R Zer so all the elements into that diagonal matrix of a square Matrix are zero then that is called as a diagonal matrix so let's see some more so for example you have uh Z 1 2 oh my gosh zero I think I I I I done wrong so 1 0 0 0 2 0 then 0 03 so this is oh I I I made the same thing again no problem in that oh this is left okay so that is the all the principal diagonal is your is your uh what do you say the the the nonzero and every every elements of except that is your not zero then that's called a diagonal matrix so I hope that you're a able to make sense out of it so the diagonal matrix should be a square Matrix and a square Matrix is nothing a matrix which said to be a square Matrix if and only if your number of rows match with the number of a columns and thus it is called the square Matrix because it looks like square but what is rectangular Matrix so one thing which I'm to mention rectangular rectangular Matrix so what is a rectangular Matrix here where your number of a rows does not matches with number of a columns oh my gosh there's a contradiction so that is nothing but called a regular Matrix sorry oh my gosh it's rectangular Matrix so let let me Define this from my school work so thanks thanks thanks my school sit down sit down yeah so uh what is rectangular Matrix a matrix is said to be a rectangular Matrix if and only if it's it's number of rows that's not match with the number of a columns so that's the rectangular Matrix so so the follow following example is a rectangular Matrix so maybe you may have 2 1 3 4 2 1 3 4 that is your number of rows is 2x 4 where 2 is not equals to 4 so that is the rectangular meter that looks like rectangle so that's why we have written the rectangular Matrix cool so let's go further into learning about scalar Matrix so what is scalar so first of all Define a scalar and then try to identify what is scal or Matrix please go ahead and write your answer let's give you give the guess guys U I'm just here to have a fun with you all so give a guess what do you mean by scal or mat because when I started first I given a very good guess and that was totally wrong because this this is like a scaler Matrix so I was little little bit okay scaler is just a number and scaler and these Matrix are are the are the aray of a numbers so how I can consider a scaler as ARS of a numbers that is the best assumption that I made at that point but no problem the scalar Matrix here's the here's your scalar Matrix in in front of you here's the scalar Matrix in front of you so you have - 7 0 0 0 - 7 0 0 0 - 7 0 0 - 7 so listen so this is called scalar Matrix this is called the scalar Matrix this is called the scalar Matrix but now you will say hey I use just now you taught the diagonal matrix in the diagonal matrix you all the elements except the principal diagonal are equal or are zero then it is a diagonal matri so here also your all the elements all the elements except your principal diagonal are zero so why not be calling at as diagonal matrix so I would ask you to have a closer look at this is and tell me what you see over here so if you if you if you if you zoom in further or or if you or if you wear your sunglasses not sunglass if you wear your goggles with minus 2.5 power you will see that your diagonal matrix uh diagonal diagonals principal diagonals scalers are all equal okay so that's what make it as a scalar mates okay so what is scalar Matrix a scalar Matrix is say sorry not a scalar a matrix is said to be an scalar Matrix if and only if if it is principal if all the elements in the principal diagonal are non are zero and principal diagonal elements are should be equals to each other okay so that that is called the scalar matrices so if all the elements in the diagonal matrix okay so all the elements in the diagonal matrix and and what are diagonal matrix diagonal matrices are the matrices where the elements except the main diagonal are zero so if all the elements in the diagonal matrix are uh EX in all the all the matrices in the diagonal matrices uh are of the of of the D is is equal means the principal principal diagonal is equal then that is called the scalar Matrix so that is a scal matrix so please see the notes in description for the F the definition whatever I'm telling so you could not write it out please see the description for the notes of whatever I'm telling okay so so so so one of the so this is your first example so let's say second example second example otk 5 0 0 0 < TK 5 0 0 0un five what is this this is a scalar Matrix now now what what what what do I tell to you is to multiply with some multiply it's with with with the some Matrix okay so multi multiply with the same Matrix multiply with the same Matrix M multiply some The Matrix and then you will getting some other result some other result but I want to tell is to have a matrix like this uh where your all the diagonals are one all the diagonals are one that is your so now multiply with any Matrix just to just to make sure that is a matrix M multiplication is defined multiply with any Matrix or a vector any Matrix or or a vector you will be getting you will be getting your your your Matrix so this is this is this this this is your scaler Matrix so let's say s as as as of now and this is your any Matrix or vector v so when you multiply s * V answer will be V means exact so is it is by multiplying by one you will get exactly so please see please do and see the experiment so when you it this when you multiply this Matrix with some other Matrix or vector you will be getting it is just like multiplying this Vector this is this is one so whenever you multiply s * V means this these types of Matrix where your principal principal diagonal is one all are one then you multiply with some Matrix or vector then that then that will yield or or result to this Vector original Vector to which you multiply that scalar Matrix to okay so that is so so so scientist have seen this y result and named this as a identity Matrix name this as identity Matrix or a unit Matrix or a unit Matrix okay so when you when you multiply this identity Matrix with any Matrix that is simply by multiplying one and you will be getting a result which is V okay so you'll be getting the same result so you have you may consider this Matrix with one and if you multiply any even 10 you will be getting your 10 as output so it is same as that identity Matrix oh my gosh that that the one who's is just flying the of pollution I am really not liking that no problem again so here over here your identity Matrix are just like a m multiplication by one so please pre to this is a wi property which we have given a new name is the Matrix or the scaler Matrix here in your principal diagonals are all one then that's uh nothing but uh identity Matrix okay let's let's go on next page the next page so let's talk about some last matri with something called a triangular Matrix triangular triangular Matrix so the Triangular Matrix are of two types so a square Matrix I'm just try to Matrix so a square Matrix is said to be a triangular Matrix if the elements if the elements if the elements Ave oh my gosh my hand ratting Ave or below the principal diagonal below the principal diagonal principal diagonal are zero okay so for example uh you have the diagonal 3 4 6 1 2 3 and z z0 okay okay so this is your oh my gosh what do I made over here 3 1 2 0 4 3 0 0 6 okay so it is telling the Triangular this this is a triangular Matrix because all the elements if the elements above means Above This is this this is your principal diagonal this is your principal diagonal okay so whatever Ave if either above or below either above or below yeah here is your or okay so either above or below either above or below either above or below um to the main principal diagonal are zero then that's called a triangular Matrix so here above is non Zer but below is zero so that's how we call it as a triangular Matrix because it forms a triangle so that's why it's called a triangular Matrix and this is called the upper triangular and here here it forms the triangle so here it forms a triangular part okay so that is the upper triangle so this this where your zeros are below the main principal diagonal so that is called the upper upper triangular Matrix upper triangular Matrix and for example you have 1 0 0 2 3 0 4 5 2 and over here this is your main diagonal okay okay and above you have zero and below so that that is the uh that is the uh lower triangular Matrix that is a lower triangular because it forms a triangle triangle low and below below of the main diagonal okay so that is the lower lower triangle Matrix and upper triangular Matrix cool so I hope that you understood so just just to re recapitulate the Triangular Matrix is said to be a triangular Matrix if the elements above that principal diagonal or below the principal diagonal are zero the the the the the the elements uh the or or the what do you say if the zeros are below the principal diagonal of the Triangular Matrix then that is called the upper triangular Matrix because it forms the triangle upper of the principal diagonal and if the in in the in the Triangular Matrix to the of your principal diagonal of if your zeros are above your principal diagonal then that's nothing called as a triang a lower triangular Matrix that is of two types cool the last thing which I'm to discuss is about symmetric Matrix okay so is about symmetric matrix it's about what do you say repeat me with me symmetric Matrix symmetric who knows so you have this so you have a and when you do this so it should be foldable so that is symmetric so this this paper is symmetric okay so the same way the the the M matri can be symmetric as well the M matrices can be symmetric as well so what is symmetric Matrix so so the definition of a symmetric Matrix definition of a symmetric Matrix if your a transpose is equals to the a a is if your a transpose is equals to the a so that's where we call that as a as a as a triangle or or or a symmetric Matrix okay so it's it's it's it's called a symmetric Matrix if your a transpose is equals to A itself okay so so for example so for example you I'm just going to take you you can think of any example this is your task but I'm just just going to take a small example of uh 2 4 69 okay so 2 469 so when you add the transpose so this is your 2x2 matrix and then if you do the transpose you'll be nothing 2649 2x two 2x two okay so that is the 2x two so over here here you follow the equality of a matrix so the you if you remember the equality of a matrix if you remember the quality of a matrix of of the matrices so if if a matrix a is one equals to Matrix B if its corresponding elements if it's corresponding elements this to this this to this this to this this to this are equal and they are of same order and they are of same size okay so they are of same size but this is equals to this okay four is not equals to 6 so here this is not a symmetric Matrix okay so this is not a symmetric Matrix so your all symmetric Matrix should be should be square Matrix to be symmetric okay but not every Square Matrix can be symmetric but but you for for being symmetric your your Matrix should be squar Matrix for for being symmetric but it is but it's not guaranteed that your every Square Matrix will be symmetric but for being metric it is you have to have a square Matrix for example you have 2 2 2 2 apply the transpose on this what it would be 2 2 2 two 2 by two 2 by two this is this is correct this is correct corresponding are also correct the size is also correct that is the symmetric Matrix that is the symmetric Matrix and I hope that you understood about the concept behind symmetric Matrix so this is all so that was we had a talk on these stuff so I hope that you like this video and I also talked a lot and sorry for the crackers please find that guy and beat him as much as you can who's fing up the crackers outside so sorry I I I included Hindi but no problem okay please feel free to sco that guy not beat him because Dali is Festival of having fun but yeah please SC that because they disturb you in studying no problem uh so let's so I I I hope that you understood and please feel free to do your home homework assignments so here is the homework assignment uh the discussion for the solutions of the program programming assignment or sorry homework assignment is being released soon in the form of video so you can assist that but I will wait for two two or 3 days and then I will release one video on uh solving these homework assignments please feel free to do this and please feel and I'm not giving these notes because these notes are already available in the description down box below in the form of some uh good good hand handwriting in the description down box below please please feel free to assess thanks for seeing this video I'll be catching up in the next video till then bye-bye have a great day [Music] bye-bye hey everyone so let's get started with a new lecture on lecture number seven which is on determinant and this is one of the one of the again I would say important concept to study because in principal comat analysis or whether you uh it it it it comes a lot in your machine Learning Journey as well as well as in deep Learning Journey because it tells you how to solve or solving the linear equations or or or or if if I talk about in terms of linear transformation it just tells you how the how the how the change in area or a volume occurs okay and and determinant is nothing when you it's nothing but you just TR you just give some Matrix and then you get one number so we'll be talking about that in detail in this session uh I I think you'll you'll get a lot from this session and and you you can make your own notes or the notes is in description un boox below either it would be updated soon but yeah uh I it is it is already been made it's just sent for processing and that it will be into your description if you like this video please be sure to subscribe this channel as well as like this video and comment because YouTube algorithm knows okay this is a good video to recommend because many many other the people say uh your channel is underrated so I want you I want this channel to be rated Channel because I work a lot on this channel Okay cool so let's get start with solving uh what is determinant so we'll we'll get onto the geometric meaning soon but uh in in determinant what you do if you know about a square Matrix if you know about a square Matrix which which we talked about and and I have told that is very important Square Matrix are very important is used extensively in linear algebra to to use this term terminology so Square Matrix is nothing where your where your number of a rows is equals to the number of a columns for example uh your Matrix a is is maybe it can be 2 2 2 two okay so this is a 2X 2 where n = 2 and M = 2 so n * n Matrix where your Square Matrix is equals to where your number of rows is equals to the number of a columns okay so that is the so this is so what what you do you take your Square Matrix and determinant takes one square Matrix where the number of rows is equals to the number of colums you write determinant of uh an A and A A should be the square Matrix a should be the square Matrix and then you get one scalar or or or a number as an output when you apply the determinant function or or or when you take out the determinant of that Matrix okay so now how this is useful we will see how do we take out the scalar a just in a second numerically but but uh but when you um how how the determinant is useful this is this is one of the most important concept to know so the determinant is useful in in solving and solving linear equation and linear equation is used very very extensively solving linear equation or maybe it can be useful in in in in in in knowing okay in knowing how linear transformation and knowing how linear how linear transformation transformation change their area or the volume okay change their area transformation change their area change their area over volume or volume okay not over it's or volume and it is also and it is also useful uh in other stuffs like uh when solving some some computationally it it it it it does reduces some computer not exactly mean uh doing efficiently not exactly I would say efficiently I would say very precisely so solving the particular linear equation and is used a lot in that so that's why we take out the determinant of a matrix and that when you take out the determinant of a matrix you simply give a squar matrix root to that determinant and then after when you take out the determinant you will get one scaler okay so this is what the this is this is this is what we use and and if you if you talk about um in machine machine learning use case so in machine learning if if you know about machine learning in machine learning you have something called as dimensionality reduction method and in and in that you take out the determinant of that co-variance Matrix so co-variance Matrix okay so when you take out the determinant of that co-variance Matrix and then you and then and and and and then go further into solving the particular problem okay so not exactly covariance yeah so you take out the termin and then you go further into uh into other stuffs like uh uh the igen vectors and igen values and they are extensively used the determinant are extensively used in the igen vectors and igen values in principal component analysis okay so I hope that this is clear why we use determinant and and and what's the determinant is now now we need to care about how do we take out the scalar value because we give a function because we just give a a square Matrix into that determinant and then we will we are going we are we are just getting a scaler as an output so how do we even do that uh so for for doing that assume that you have a matrix a you have a matrix a which is nothing but 2x two so I'm just going to write um a b c and d okay so you have a matrix a b c d which is a 2X 2 Matrix so when you take out the determinant of that Matrix a which is nothing but which is nothing but so a means you take out the product of the diagonals you take out the product of the diagonals a minus BC a D minus BC so for example you have a matrix uh 2 3 4 6 and then you want to take out the Matrix the determinant of that Matrix 2x2 matrix which is nothing but 2 * 6 2 * 6 - 3 3 3 * 4 3 * 4 which is nothing 6 6 2 12 - 3 42 that will be nothing but zero zero is the answer or determinant of this Matrix okay so the terminant of a matrix can be zero we have we don't have any conditions but yeah the determinant of this Matrix is zero okay so this is how you take out the determinant of a matrix geometrically speaking okay so one thing that I want to highlight over here let's say for for example uh what does it mean geometrically what does it mean geometrically so so let's uh let me make one more page so that I could explain you what does it mean geometrically speaking what does it mean geometrically speaking either I could just go on some website to mean to mean what is actually trying to tell so let's go on one website let's go on one website which I want to show you all is this one okay so assume that over here of over here you have let me choose my black color okay here it is so you have um a matrix a matrix a b c d okay you want to take out the determinant of this so this this is this is what you take out so for taking out the determinant you just write either in this A B C D giving a pipelines like this okay or or you write determinant of this uh a a matrix and this a matrix is either uh a b c d like this okay so this is the notation for sooning that you want to take out the determinant of this Matrix okay that pipeline that big big pipeline okay pipe uh line okay now over here your a is 1 your B is zero your C is zero and your D is one okay you want to take out the determinant of this you want to take out the determinant of this you want to take out the determinant of this so how do you take out so what does it mean geometrically speaking so geometrically what it's trying to tell is when you plot this Matrix over here first of all you take this and then you go over here so this is nothing but the determinant of a 2X two Matrix is the area of a parallelogram with the column vectors AC and BD okay so this is the the the determinant is nothing but the area of this parallelogram of this parallelogram where the column vectors are AC and BD okay so when you when you plot the 2x2 matrix which which looks like this and and and this the the the determinant which means jum Al speaking is nothing but area of that parallelogram which formed by joining everything and then and that area of that parallelogram is nothing but determinant of that Matrix okay this is what does it mean geometrically speaking uh I would ask you to watch one video on three blue one brown to see how how is shown geometrically but yeah uh the the det terminal is nothing but the area of that parallelogram whatever forms so for example you your par so let me reduce the a a bit and then let me do something with this I don't know well how it is working yeah so let me do something like this and let me increase the area okay let me increase the B okay here it is so when you have the column Vector when you have a column Vector at 0.86 and zero okay and then you have another column vector which is 0.52 and the the parallelogram is formed is nothing but your favorite the determinant okay so this is what the determinant means and you can play with it by just going to demonstration wallframe and this with this website so let's go on the 3D view so how does it look 3D so 3D is nothing but area area of that parallel Zoid okay so if you just see over here the area of the paraloid is is nothing but a determinant we'll see how to solve how how to solve this deter this determinant one okay we'll see how to solve um three for the how to take out the determinant of a 3X3 Matrix and we will also see how to take out the determinant of a n byn Matrix okay so it's a it's a bit hectic task but we will try to do it so this is this is what the geometrically means and for 2D the area of a parallelogram and for 3D area of a parallel Zoid okay which you can see from the diagrams which are shown over here so if you just if if I could zoom in I can't zoom in but yeah I can just show you this is this this is what you have your uh 3x3 Matrix and then you this is the paraloid which is formed and then when you try to take out the determinant of this is nothing but the area of this parallel Zoid okay so this is what it means and the determinant geometrically is nothing but the area of a parallelogram or parallel joid in 3D dimension okay so this is this is what you need in in a geometric intuition just just just to make sure that what the geometrical it it means okay so now let's see now one of one of the important thing which I want to show you up is is is we have seen we have seen how do we take out the determinant of a 2X two Matrix so the determinant so here is your a here's is your here is your a and you have and then you want to take out the determinant of this A B C D and I'm just writing pipe to denote okay this is a determinant so when you take when you try to take out the determinant of this so it's nothing but equals to uh uh a a minus oh my gosh it's a D minus BC that's uh then when when you take up that's a uh simple scalar which is e not exactly that not 3+ 3.71 1 it's e okay so let's let's give it any scaler which is e okay so this is this is what it means in 2x2 matrix I'm talking specifically 2x2 matrix now now let's talk about how do we take out the determinant of uh 3x3 Matrix so determinant determinant determinant of 3x3 Matrix 3x3 Matrix so how do we even approach we taking out so you have want to take out the determinant of a b c d e f g i okay so G h i is this is your Matrix this is the determinant of this Matrix okay so how do you take out how how do take a determinant of this Matrix and of course your it should be a one scalar okay it should be one scaler or a number or number okay so how do we take out the determinant so can't we do a * a * e * I and then it will not work this is this is not you can you you can just guess how do we do it just try and commment and maybe I can just see and be a bit funny in job so please be sure to write it and I will try to see what you write it okay so so let's start approaching how do we even approach this problem so what we do we simply so so what we do just just make sure that first of all we go to the a11 okay so me first element in that Matrix and then what we do we simply leave this uh column and this row and write a minor Matrix or a submatrix of that of of that uh big Matrix or you can say that we take out the minor of this Matrix how do we take out the minor of this Matrix you simply when for for example you choose this number okay so what you do you you leave this column and you leave this row and then you write uh and then what you do you take out the minor and then you take out the determiner determinant of that by multiplying by a okay so the first element is this and then you have e f h i we left this column and this row and then we write e f hi okay we want to take out the determinant of EF HR okay now what you do now what you do here is your plus sign now it will be a minus sign over here okay you go to the B you leave this column and you leave this row okay which is nothing but b and d f g i DFG because we left this column this row and this column just d f g i okay and then here is your minus then here will be plus plus uh you write C now we left this column this this this column and the first row which is which will be left the determinant of D GH okay and then we have convert now these are called a minor or a submatrix submatrix or the minor of our Matrix a these These are called the min minor these are called nothing but the minor these are nothing but called the minor minor of our Matrix of our Matrix a okay so when you try to now it is very easy a * a * uh EI now you can just apply your 2 x two a EI and FH EI minus FH okay minus b d i FH d i minus FH okay plus C and then you have DH EG okay DH minus EG okay and then you'll be left with some scalar and then you can simply do do this thing and then you simply multiply with this and then you do do some calculation and then you'll be getting your output as maybe some some scalar some scalar value okay so let's see one of one of the one of the one of the problem or or the stuff to to see how how it looks like Okay so let's let's assume that you have a a matrix or 3x3 Matrix so here's a question for you okay maybe you can try try to approach it uh the you want to take over the determinant of I'm writing this pipe that denotes that you want to take out the determinant of that uh for example 0 1 2 uh 1 2 0 uh let's let write 1 1 0 okay just a random random I'll be walking you through it so take out the determinant of this this is a 3X3 Matrix try to take out the determinant of this so how do we take out so first of all we go through the first element and then what we do we take out the minor of this Matrix so the minor so we leave this column and this row so we'll be left with zero and then we and then we write out minor and then we take out the determinant of our sub Matrix okay plus now no no no it will be not plus over here it will be minus because here is our plus minus okay one you leave this column and this row which will 1 0 1 0 okay so 1 1 0 1 0 okay and then you write + 2 and then you have uh you leave one to one one okay you leave this column and this row okay you leave this column and this row you'll be left with one 2 1 1 okay and then you do the sum and then you do the sum so and then you take and then what you do you try to take out zero 2 * 0 which is 0 - 0 okay Min - uh 1 * 1 * 0 of course 0 and 1 * 0 0 okay plus two uh 2 okay 1 * 1 1 2 * 1 2 okay then you'll be left with of course zero then it will be done then done it will be also 1 * Z which is nothing but zero okay it'll be left it 2 * - 2 2 * -2 that that will be -4 okay which will which is is your determinant of this Matrix so Min -4 is your determinant of this Matrix which you are seeing over here so sorry here is 1 - 2 it's not it's it's simply min-1 so 2 * -2 is the determinant of this Matrix so I'll be so here you got the determinant of this Matrix which is nothing but min-2 okay so this is this is how you take out the determinant of a 3X3 Matrix as well so there are there are some problem for you to work on so I'm just just going to write it out so there's one problem which which which you can approach okay so you want to take out the determinant of uh 371 -4 please answer the HW please answer in the comment box it's just a quiz which you will see in your attendance as well okay so this is what the determinant of that Matrix of the 2x2 matrix or 3x3 Matrix we'll try to solve the determinant of a 3X3 Matrix using lianes formula or the rule of surus okay so which is very good formula to work on so we'll see that but before that I want to highlight some of the properties some of the properties of that properties properties of determinant Matrix of determinant of a matrix determinant of a matrix the first first one is the first one is the first one is for example you want to take out the determinant of this Matrix for example you to take out the determinant of 1 0 uh 0 1 okay so try try try to take out the determinant of this 1 * 1 which is 1 uh and then and then minus 0 okay what what it will it with one and can you identify this is an identity Matrix even if you have uh three three identity Matrix then that will be nothing but that will be nothing but one so whenever you have identity Matrix whenever you have if if if it is if it is identity Matrix if it is ENT identity Matrix identity Matrix then then then the then the then the determinant of that identity Matrix will be one okay this is this is one of the property second property if the if the rows are the same okay so for example if the rows are the same are the same for for example a a b b okay a a b b then a minus ba will be nothing but zero okay so this is another property third property is you have a scaler multiply it with some uh a and you have another scale c and b d okay so what it will be it it just makes sense r a D minus r CD or r r CB okay you can write write it down like this and then it's it's nothing it's nothing you just you just take that out of out of okay you just uh take take that as a common r a D minus BC okay so we can write this as a we can write this we can write this as a r * a b c d either we can write it now so we proved it so it is just equivalent you can write this so for it will be easily for us to solve okay so it is so it is R times as either it is same as over here so we a D minus BC so it's just equivalent to that so this is another property which you see a lot in while taking of the determinant of a matrix okay so this is these are the some of some of the properties which I want to highlight in front of you so now let's go on to the another stuff is how do we take out the determinant how do we even bother taking out the determinant determinant determinant of 3x3 Matrix so you can just say okay I'm just going to just going to take out the minor of the sub Matrix of the Matrix and then I will do that so here's another another trick which is called the rule of surus I think it's it's it's not a I would say uh okay it's a good technique but okay you can try it out but eventually I like that my Approach but yeah it is very very straightforward approach which I'm going to tell over here okay so assume that you have a matrix M that you have a matrix M Okay so a11 1 a12 a13 okay a21 a22 a23 a23 a31 a32 a33 okay so you have this 3x3 Matrix now when you wanted to take out the determinant determinant of this Matrix M so how do we even bother doing that so for for for for so you can use the rule of suus rule of suus I think that funny name he has but yeah again I'm no no want to comment on on his name he's a again a great people okay so not I'm not even a one 1% of these people so these are amazing people who give a lot to the world so I think about I'm no one to say about but yeah amazing name so what you do you take out the determinant so you want to take out the determinant of a11 a12 a13 okay so this Matrix I'm just writing this Matrix a21 a22 a23 okay a31 a32 a33 okay and then what you do you take out the first two column and write it in another format like this a11 a12 A2 1 this is a trick for solving a 3X3 Matrix a22 and a 31 okay so a31 and a32 so this is a22 okay it's say a21 okay so this is what you now you write this now what you do now what you do so what you do you you simply take out the product you simply take out the product like this the first diagonal okay so this is the diagonal so what you do a11 a 22 a33 okay a11 a22 a33 plus plus a12 a22 a23 okay A2 3 and a31 so what you do you take out the product of these three you take out the product of these three so A1 2 a uh uh 23 okay a 23 uh and a 31 okay now what you do you simply uh do this simply multiply the next next diagonally okay so plus plus a13 a21 okay I'm I'm I'm I'm doing a bit messy so let me do a31 uh if I'm not wrong a13 a a a13 a21 and A3 2 okay A a A1 3 A2 1 a32 now you are done with this now what you do now what you do you now go from bottom to top here you are going from top to bottom now you'll go from bottom to top by changing the sign now okay now you go from bottom to top so here's how you do here's how you go further okay so so the way you go is you have a31 so from here so from here a 3 1 okay and then a22 and then you go to a a13 so now you start going at this side like like this okay so a31 I'm just going to write a31 a22 a22 and then a13 a13 okay and then what you do and then plus no uh you you minus because you change you go from bottom to top so here you minus it now now you go at this one now you do this and then a32 okay this one this one and this one a32 * a23 yeah if I'm it's a it's a a23 if I'm not wrong yeah a23 and a a11 a11 okay now minus now this is done now you go at last one which is this one a33 a21 a12 okay and here's how you take out the determinant of a matrix using suus rule okay or a rule of suus okay so and then you'll be getting after after doing this all those stuffs you can just cancel it out something if it is so you just you just do do the computation then here's your Matrix and this is just a scalar number or other stuffs so here's the rule of suru so here's how you do you do simply you do you do write the first two column uh at at the side so that it could be easily so uh so what you do you simply take of the product from top to bottom for the first three and then you take out the bottom to top for the second three starting from the last okay so here's here's here's what the full the rule of sarus means here here's how you take out the determinant of that Matrix like this okay so now I'm going to talk about is uh you can see the Wikipedia Pages for a libanese rule because they write a very very kind of libanese stuff so you can just go there and see more see more about this rule okay so the next thing which I'm going to talk about is the next thing which I'm going to talk about is how do we take out the determinant how do we take out even B of taking out the determinant of n by n Matrix the determinant of n by n Matrix so how do we even take out that and how do we even bother taking out that okay so here's so I'm just going to write the N by n Matrix as I'm just going to Rite the N by n Matrix or let's start with a particular example let's start with a particular example so it would to totally make sense okay so let's let's start with a particular example and then at last we'll just write a definition and then we'll end this video okay the example is bit long so I'm just going to maintain my handwriting so the example is you want to take out the determinant you want to take out the determinant of a 4x4 Matrix 1 2 3 oh my gosh three 4 okay 6 6 6 9 2 1 and then you have a 4 9 2 1 and then you have a 0 1 1 one okay so here's here's your determinant of this Matrix so to take out the determinant of this Matrix so how do we even approach taking out the determinant of this Matrix so how do we take out the determinant of this so for taking out the determinant of this so for taking out the determinant of this for for taking out the determinant of this you take out you take out you first of all take a the minor or the submatrix of this okay so you you go to the First Column first element and then you leave this column and this row and then write the sub Matrix so you just one and then you take out the determinant of 9 2 1 9 21 1 1 1 okay this is plus sign so it will be minus sign now you go at this uh take take out the sub Matrix leaving this row this column and this row so it would be two first of all you take product it so you multiply with that 2 2 * uh the determinant of 6 2 1 6 2 1 42 1 and 011 okay and then you uh change the Sign Plus and then go through with three and then you leave this column and this row so it will be nothing but uh three and then you have 621 I also just write okay 66 6 91 I I just leave it so I'm three 691 691 uh 4 4 91 4 91 and 01 1 okay 01 1 now the last one is there four okay so there is four and then you take out the terminant of leaving all the all the uh column one and then row 692 692 492 and uh okay I think it's wrong 492 and 0 1 1 okay so these are the sub Matrix of that Matrix let's name it as an n M okay so this is a matrix M and then you have to take a determinat of that Matrix M so here's the 4 4x4 now you do this now you have this now what you do now here you convert it to 3x3 determinant now what you do you convert that to a 2x2 here's how you do so you you don't want to use a suus rule because I I eventually don't like that rule it's very hectic rule sometimes it maybe cause you error but no problem in that so here's how you do it so first of all what you do so first of all what you do you you simply uh multiply uh one okay you simply go ahead and take take your uh one as a so if you can see I just want to take that one as an uh uh this one and then what I and then I go and approaching this so here is your nine so first first of all go at this element take out nine you want to take out the sub Matrix of this Matrix so nine and then you take the termin of 2 1 1 1 so what how this came 2 1 uh you leave this this row and this column 21 1 1 okay now you simply change the sign minus okay you make sure that t you're doing only doing for this you're only doing for this we'll come to this later on but we are only doing for for this a21 okay minus now now we go to this two now we go this two we leave this column and this row which is 91 1 1 okay so which is a what happened yeah so which is nothing but uh what do you say uh two because here is our two leaving this row this this this column and this row 9111 okay uh did the determinant of 91 1 1 okay and then what you do plus now you change the sign and then you go at last 1 leaving this 9 to11 okay so the one and the 921 1 okay so you take out that okay now this was plus now you make it minus okay now here is your two so I will take that outside I will take take that outside okay and then I will just go ahead into solving this so you take this take this as a now you take out the minor of this Matrix or the sub Matrix of this Matrix so here's how you do it so you simply add it minus so here is minus 6 uh 21 1 1 so here here's how you go with this you ignore this column and this row 211 1 now you go to this you ignore this 41 0 41 0 1 and then you go over here ignore this 42 01 okay so this is how I'm I'm I'm going to write minus 2 okay because you go over here 2 over minus 2 CH changing the sign take out the determinant of 4 1 0 1 4 1 01 okay and then you simply plus um now you change the sign you go one go to go to one 42 0 1 42 0 1 uh + 1 uh 4 2 0 1 okay and then what you do and oh my gosh yeah so then what do you do now you now you converted that 3x3 Matrix for this one and for this one now you go to this one okay by changing the sign plus plus and then you go and then you write separate three now you write separate three and then you take out the first one six okay so you leave this SC column and this row which is nothing but uh six and then 9111 which is the sum Matrix of that Matrix Min - 9 because this plus 9 and how how how came you go with this column leaving this column and leaving this row 41 01 which is nothing but 4101 okay plus changing the sign 149 01 how this 4901 came is you have this you leave this and this you leave this column and this the row which is 4901 okay so this is how you came it and then and then you're done okay now what do you do you you do for the last one you do for the last one this because you you done for this you're done for this you're done for this now you converted that to a 2x2 MRI which is easily deter which which we can easily take out the determinant now you go to this okay so here's how you do it so Min - 4 okay and then what do you do and then what do you do you leave this column and this row taking out the first element so six take the determinant of I was I think it's where it was 9 9211 9211 9211 minus minus 9 okay so over here it was leaving this the SE going to this and leaving this column and this row which is 4201 I I I think about it yeah that's 4201 4201 and then you change the sign plus 2 49 I think it's it's it's more about you leave you go over here 4901 okay that is 4901 okay now you are done now this is what you have written so you converted the first Matrix the first Matrix this one this one and this one as well uh into a minor Matrix which is 2x2 determinant so you can easily take out and then do the product and then you take out okay so let's do over here if if I have a chance to do over here but no problem I will do over here okay so here's how you do it here's how you do it so for doing it first of all you have the one available which is over here you have the one available which is over here what do you do you simply 9 - 16 + 7 how how how we came so you have the particularly nine times because of course you want to always want to multiply it out okay so I think about this is you have uh if if you go over here 2 * 1 okay and 1 * 1 so 2 2 2 * 1 how much 2 * 1 how much it would be uh 1 - 1 I would say uh 2 - 1 which is 1 so it will be 9 okay minus minus over here uh 9 9 * 1 which is 9 - 1 which is 8 16 - 16 done and then you have 9 * 1 and then minus 2 * 1 so 9 - 2 which is 7 okay so here's how it came okay and then you then this the left minus two now you go on second one two over here so when you when you take a two * 1 how much 2 * 1 how much 2 * 1 2 - 1 okay that is 6 over here here so we write uh 6 - 8 + 4 okay so here's how you do so it is 6us 4 * 1 of course 4 * 2 it's 0 * 1 of course 0 so 4 * 1 4 * - 2 which is - 8 which which you have written over here okay then you go over here 4 * 1 how much 4 * 1 4 and then you four so here here is a plus 4 now now you go to the next + three because you go over here now 9 * 1 how much 9 * 1 how much uh 9 * 1 9 of course - 1 uh which is nothing but 8 8 * 6 48 so you have 48 - 36 + 4 okay so here's our 48 - 9 uh so 4 * 1 4 so 4 9 49 36 because this 2 * 0 is 0 then you go over here then you have this 9 9 9 * 0 is of course 0 4 * 1 so 4 2 are 8 eight so over here uh I think I'm wrong over here um you have this 4 * 1 4 91 0 0 so 4 uh it's it's it's it's four okay so it's four it's it it it it it should be eight okay it should be 8 which is nothing but what you do you 4 * 1 4 and then you simply multiply with two which is nothing but 8 okay now you simply - - 4 - 4 uh 28 - 30 6 + 88 so how you take out 9 * 1 how much 9 - 2 7 7 7 6 7 76 how much uh oh oops it's it's uh it should be 9 * 1 - 2 987 76 42 okay so which will be nothing but 42 what the hell i' written over here so I have to just couple it out so I have to just return it it will be nothing but 42 okay minus minus minus uh minus 4 * 1 of course 4 9 are 36 plus 2 4 4 1 are 4 4 2 8 okay so here's how you do it now you simply multiply with this and then first of all do the calculation do the calculation then you'll be getting one a scaler as your output so please feel free to put your answer in the description box below I know the answer but yeah I want to leave it to you to do the rest of the calculation I have done a lot so here's how you take a the determinant of n byn Matrix and how you do it and how you do it it's very very easy you just uh keep converting that to a lower or a sub Matrix and then you and then after that you are done okay so here's how you do it so you you define one you define one sub Matrix AIG J you define one sub Matrix AIG J which is nothing but the Matrix the n n minus 1 * n-1 Matrix n-1 n-1 Matrix if you ignore ignore the I row and I column if you ignore if you ignore the I row which which you are doing I row and J column which which you were doing okay that is your new Matrix which which we were forming that is a recursive this is a recursive you can write a Python program for write a recursive solution for this okay so you were writing a one one and then you were adding the determinant of that subm Matrix and then you are doing so and so on so on that is a recursive solution so you writing the recursive recursive stuff so we we we already written a lot so I hope that you understood for formal definition which you can see yeah we have gone through one of one of the example which is very very much important for us to know okay so I hope that you will uh get a lot from this video and determine it I hope that concept is clear with my examples and I also hope that you enjoyed this video I think I have to wrap up with wrap up with this video I'll be catching up your next video till then bye-bye have a have a great day meet you in the next [Music] lecture okay everyone let's get started with another lecture I know it's bit L late lecture but I apologize for that I'll be I'll I'll be making making sure that I'll be providing you around three to four videos this week so I already provided two videos now I think about two of three videos will provided more this week till your next uh assignment or homework assignment and please make sure that your homework assignment is released if we find any student who are not active we will remove them from our LMS because this is an opportunity which are given for free for others to learn because if the people do not take this opportunity we are going to drop that student so uh we we we highly recommend to to to to be active on LMS please do your assignments please attend your attendance and every stuff okay so please please go there and Mark your attendance and as well as uh complete your assignments even if you complete around out of 25 questions you you have to complete around 20 questions you can complete 20 questions write right in Notebook and then give it to us okay so you your programming assignment sorry the homework assignment will be will be evaluated and then that will be uh till the end end of the course and if we do not find you active in the course we will drop you out okay so this is one of those announcement that our team has told me to give me to give it to give it to you all through Me Okay cool so so what's the mod of this lecture the M of this lecture to talk about the co-actor the minor and educate or the adjoint and invoce of a matrix so these are very very correlated these are very very correlated uh for for taking all the co-actor you need mind and for taking out the minor you need determinant and for taking out the aducate you need co-actor and taking out the inverse you need ugate okay so again I'm explaining for for for taking out the co-actor of a matrix for taking out the co-actor of a matrix you need minor and for taking of the minor you need determinant and and and and after after you take out and for taking out the adjugate for taking out the adjugate for taking out the adjugate for taking out Agate you need co-actor and then for taking out the inverse of a matrix you need aducate okay so these are very very correlated and they are heavily used for many this inverse of a matrix because they are so correlated so I thought okay let's let's start with this video so that everyone knows about co-actor how the N inverse of a matrix is calculated because it is extensively used in the industry okay uh mainly in linear algebra there's not too much use in machine learning some sometimes us machine learning but it's very very good to know about these stuffs okay so first of all how do you be now let's let's let's go ahead and talking about the first two stuff which is which is a minor of a matrix or the co-actor of a matrix so let's start with a minor let's start with a minor minor of a matrix so so so a minor of a matrix a as as if if you remember the minor the a minor of a of of a Matrix a a minor of a matrix a matrix a is the determinant of the small same some some smaller um Square Matrix um a minor of a matrix a is the determinant is the determinant is the determinant of some of some smaller of some smaller squar Matrix as you remember remember that what we do what we do if we were removing the column and the row for that for that point or the element and then we were we were obtaining a sum Matrix in the determinant and that's actually when you when you take out the determinant of of that some Sub sub Matrix that's actually the minor we we will see see one example just just in a second for example for example let's take an example that you have the following that you have the following uh Matrix okay you want to take out the determinant of this Matrix or the minor of this Matrix you want to take out the minor of this Matrix so for example you told okay you want to take the minor of Matrix M for I row and J column okay that so you take out so you for example you choose okay you want to remove the second row uh and the third column okay so you are saying 2 3 so you to you want to take out the minor of a matrix given I = 2 and J = to 3 so what what it will do it will it will leave second row and it will leave the third column okay so the minor of this Matrix a will be left with the the the sub Matrix will be left with 14 - 1 9 as you all are knowing okay so now when you take out the determinant of this Matrix so 1 minus uh this is first of all for taking on the determinant of a 2x2 matrix what we do we simply what we do we simply uh multiply the DI diagonals and then sub subtract it so 1 uh my time uh 9 okay minus uh of course uh sorry this minus and minus 4 okay so that will be left with 9 + 4 which is nothing but 13 so 13 is a minor of a Matrix given I = 2 and J = 3 so what does it mean we leave the second row and third column or we delete the second row and third column to obtain the sub Matrix so that is the minor of that Matrix okay so again explaining what we do we simply remove just one row and one column and we take what row you want to remove and what column you want to remove by the user I and J you you remove one row and one column from the square Matrix and make sure that is a square Matrix what is the square Matrix the square Matrix is the one where the number of rows the number of rows matches with the number number of columns okay so the number of a rows matches with the number of a columns and over here when you take out the minor of this Matrix a given you remove one row and one column so you rem the remove the second row and the third column for example if you want to take one one so the minor of this will be you have you want to remove the first row and First Column the minor will be determinant of that uh 3 0 so sorry it will be it will be I think 0 5 91 so take the determinant of this so 0 0 * 11 0 - 9 * 5 45 what it will be it will be 45 or - 45 okay so that will 45 around according to this okay so that's that will minus 45 so that's how what this minor tells you minor tells you okay you want to take out you want to remove that column I you want to remove that I row and J column and then write write down the Matrix and then take out the determinant of that submatrix and whatever the determinant will be that will be your minor of that Matrix okay so this is this is what it's trying to tell you so over here oh my God what is this where is where is my pen so this is what it'sing trying to tell you over here okay so what does minor means minor of a the a minor a is the determinant of some smaller Matrix some a smaller Matrix okay so a minor a matrix a is the determinant of some smaller Matrix for for example which you're seeing over here and then you so how do you take out you remove one row and one column and ride the rest of the Matrix element into a new Matrix and then you take out the determinant of that Matrix and the whatever the determinant a scalar value that scalar value will be uh the minor of that Matrix okay so this is how you calculate the minor of the Matrix so so so so so using minor of a matrix you can calculate the co co-actor of a matrix using minor of a matrix you want to calculate the co-actor of a matrix so why do we you why do we need to calculate the minor for Matrix to calculate the co-actor you we need uh we need to calculate the minor so what does it mean so why do we even care about co-actor why do we need co-actor So Co co-actor is required for co-actor is required for computing determinants Computing Computing high level determinant or larger determinants okay Computing larger determinants or determinants and taking out and in taking out in taking out the inverse of a matrix indirectly inverse of Matrix indirectly as it is not used directly over there you want to take out the adjugate of that for taking out the inverse but adjugate uses adjugate us a CO co-actor and co-actor is being indirectly contributing to taking out the inverse of a matrix we'll see the inverse of Matrix just we will visit after some slides okay so this is this is what it means to be minor so let's talk about the co-actor of a matrix let's talk about the co-actor of a matrix so what is the co-actor co-actor is calculated first of all we need to calculate the the the minor so for example you have the following uh Matrix you have the following Matrix you have the following Matrix 147 305 -1 911 okay so this is this is your this is your um this is this is your 3x3 Square Matrix you take out the minor of this Matrix you take out the minor of this Matrix given I to be 2 and J to be 3 okay so you want to take out the determinant of the sub Matrix of the sub Matrix of the sub Matrix so for example I'm just taking one example given I = 2 and G = 3 so what you do you leave the second row and third column the second row and third column so left with 1 4 - 1 9 so this is and then when when you take out the determinant determinant of M2 3 that will be what 9 - 9 * 1 9 - um - 4 that will be nothing but 13 and as we shown 9 first of all the product of the diagonals and subtract it minus -4 so that will be what 13 okay so that is the minor of that Matrix now when you take out the minor of the Matrix how to calculate the co-actor of a matrix so for calculating the co-actor of a matrix I and j i and J and make sure that I I I matches with the minor minor of that and J matches with the minor of uh or uh whatever the uh J over here so these these two should match equals to -1 the power of I + J * the minor I and J okay so this is how you calculate so for example for this example let's calculate the co-actor so how do how how do we calculate the co-actor so the here I is 2 3 J is 3 is which which is nothing but equals to -1 -1 2 + 3 * 13 okay so whatever the value will be over over here we don't care of that it will be minus one because it's five mean min-1 to ^ 5 what it will be -1 of course because if it if it is 6 then it will be 1 so it is -1 * 13 so the output will be the the answer will be Min -3 will be the co-actor of this Matrix given I to be two and J to be three okay of that minor okay of that of the two three ENT three okay so what how you write the co-actor the co-actor of 23 entry where two is the row and J is the uh two is the row and three is the column entry is min -3 okay so this is how you C calculate the the co-actor of a particular Matrix okay so let's see one more example to make intuitive sense to you so that it would not left in sense oh my God what is happening with with some example so let's take an example that you have 2 4 6 21 2 1 one1 so this is your Matrix so you select okay you are selecting the third row you're selecting the third row you to take out first of all minor so going to take out the third row and the first column okay so third row and the First Column okay so you to take out the minor of that so first of all you leave the third row and the First Column so the minor will be the minor will be the deter the The subm Matrix will be at what uh 4 61 2 and when you try to take out the determinant of this what it will be 4 * 2 - 6 * 1 which is nothing but 8 - 6 which is nothing but 2 okay so the minor of 3A 1 entry is nothing but equals to 2 okay so after after you calculate the minor if you calculate the minor you wanted to calculate the the co-actor because you want to calculate the co-actor so for calculating the co-actor you need a minor so you taken out the minor now when you calculate the co-actor of 2x3 entry of the two two sorry it's 3x 1 now it's 3x 1 3x1 entry which will be what uh which should what so c i j so formula is c i the co-actor of three I and J entry is nothing but -1 to the power of I + j i + J times the minor Matrix I and G so in this example 2 3 = -1 2 + 3 * 2 * 2 so what it will be it will -1 to the^ of 5 * 2 okay so that will -1 time 2 which will nothing but minus 2 is your co-actor of that Matrix okay so the co-actor of that Matrix for that ENT 3 3 3A 1 is -2 so this is how you calculate the co-actor of a matrix so I'm just just going to ride the steps for you to calculate the co-actor of a matrix so first of all you take out you take out you take out the minor you take out the minor take out the minor m i j how you take out the minor so first of all you take out the sub Matrix so you remove the one row I throw I'll just remove I just going to write remove I row remove I row and J column okay and then whatever the sub Matrix would be left and then sub Matrix sub Matrix and whatever the sub Matrix take take out the determinant of that sub Matrix okay that will be a minor after you take out now you can simply calculate the co factor which will be nothing but C J which is nothing but -1 the power of y + J * m j okay so this is how you calculate the co-actor of a matrix c i j for that I entry okay so I hope that you understood what's the co-actor and what's the com Miner of that so some some of the applications so some of the application so here over here what you're trying to actually do is to uh so we can write it's basically this Co co-actors are basically used in prominently in lapl formulas for for the expansion of the larger determinants okay as as we have already seen so the formula of determinant of a which is nothing but I = to 1 all the way around to the n a i j a i j - 1 I plus j m i j okay so this is for taking of the determinant as as I told you one of the most best application of uh of the co-actors it is used in is a laplus formula for taking out the determinant of larger determinant so this is the formula for taking out the determinant of a by using the co-actor so I = to 1 all the d m a i j times uh uh times of course minus one this is the co-actor of your Matrix and this is a minor of that okay for that to I I and J entry now so we had seen the co co-actor we have seen the co-actor and we have also seen the minor and we have talked a lot about determinant and we talked a lot about determinant now let's talk about now let's talk about uh the Agate of a matrix so the adjugate of a matrix or some sometimes call it as a adjoint of a matrix so adjugate Matrix okay so we have to take out the aggate of a matrix so let's write it out so when how we take the adjugate for Matrix so first of all what is the adjugate of for Matrix so the adjugate of a matrix or the or sometimes there are lots of names sometimes we call it as the adjugate sometimes we call it is a classical adjoint classical adjoint and a lot more okay so this is a classical adjoint we also call it as that so how do we take out the adjugate of a matrix is nothing but the transp of its Co transpose transpose of its co-actor Matrix co-actor Matrix so Agate of a matrix nothing but the transpose of its co-actor Matrix okay so here's how you define it so the aggate aggate of a matrix a is nothing but transpose of that co-actor Matrix okay so this is a co-actor co-actor matrix which you taken out for every for every I and J in your Matrix okay so Agate is what is the transpose of that co-actor Matrix so here are some properties so over here the adjugate adjugate of a is nothing but C transpose and it is nothing but uh minus one of course I'm writing the formula I + j i + J time M this is this is the formula for Cal calculating okay and and I and J should go from for all the elements so I and J okay so this is the I and J and this is how you take out the aggate of a matrix this is the definition of the aggate of a matrix okay one of the property is a times the inverse of the Matrix It it means you have a matrix a and you have a matrix inverse of that so the output will be of always the identity Matrix when you multiply the a matrix and the inverse of that Matrix so what is inverse of a matrix so here's how we deal with it so inverse of Matrix of inverse of a matrix is nothing but the AI how you C calculate it if you know about 1 / a this is this this is what we write but here's how we do that so 1 over the determinant of a one over the determinant of a times the Agate of a times the Agate of a and how do we calculate the Agate of a we calculate th