Espaces vectoriels et sous espaces vectoriels.

Key Takeaways

Hello everyone, welcome to Educational. Today we're going to look at the second part of the video on vector spaces. The idea is for you to really understand that to show that the structure we're working with is a vector space structure, in practice, in exercises, what we do is show that the structure has a subspace structure. So basically, when you're given your structure to study, you try to find a larger structure and you try to say that the structure you're going to work with is a subspace of something larger, quite simply. And what's good is that showing that something is a subspace is very straightforward; there are three things to show. Whereas showing that something is a vector space (here, refer to the video I made on green spaces), there are many things to demonstrate. So the idea is very simple: if you're told to study this, good, you say, but this is included in these, this, I know is a vector space, so mass, very good, so f The thing here that I need to study is that I just have to show that f is a subspace of this vector space, quite simply. So, as soon as you work with something, try to see if it isn't contained within something larger, which would be a vector space. What is a subspace of a vector space? Well, it's very simple. Here you have your mathematical objects— I don't know what they might be, but they are actually vectors since we are working in a vector space. So yes, if you have your vectors and we tell you, " What about f?" you said. But if, in reality, f has the same rules, in quotes, a game will explain it to you so that we understand it well, if, in quotes, f has the same rules of life as the vector space above it, well, in reality, f also has a current aspect. In short, we'll say that it's a subspace of the vector space. So, I'm going to call it... to keep it simple, now I'll explain this point a little more: it's a vector space. We're going to... Realizing this requires you to study, for example, f. Obviously, the laws are the same (see points). There are two things to consider: first, to show that I've done a subspace ( 2nd year), it's necessary that when you take a vector, a vector (f1) goes into f. When you add them, you stay in the subspace; you don't leave it. So, if x belongs to f, y belongs to f - x, and y must belong to l. If you take any real number and multiply it by x, then x must also remain in the subspace. So, if I take an x ​​in the subspace, in the subspace, it takes x into the subspace f - x, which means x from f. If that's the case, basically, when you take a vector x, a scalar, you must stay in the subspace. And secondly, very importantly, the vector must be zero. I can add a small arrow to teach you more. The vector must be zero. The reader is zero for them. This vector must also be zero. The vectors of f must be zero. Indeed, remember that a zero vector, like in high school, is equal to a zero vector plus a zero vector. This rule has been lifted. The zero vector must also apply to these vectors. If you manage to demonstrate this, then you have to create a vector space. f + points, therefore f + points, has the same laws as f + points. It is a set that is included in e. If you have f, it must obviously be non- empty. Well, if you manage to demonstrate this, it's a vector space. It's relatively simple. So, the structure you have is a vector space structure. You have demonstrated that the space you are working with is a vector space. Thank you, see you soon on educational.