![]() So if I take my matrix A, which I've expressed here in So, another way of thinking about it is, well if I take my matrix A, and I multiply it by sum vector x, that's a member of this null space, I'm going to get the zero vector. Let me use a different letter, if this was m by, I don't know, seven, then this would be R seven, Vector multiplication, this has to be an n by one, an n by one, vector, and so it's gonna have n components, so it's gonna be a member of R n. The matrix multiplication work or you could say the matrix Multiplication to work, for this to be, if this is mīy n, let me write this down, if this is m by n, well in order just to make ![]() Have to be a member of R n? Well just for the matrix On why I'm saying R n, in a second, such that, such that, if I take my matrix A, if I take my matrix A, and multiply it by one of those x's, by one of those x's, I'm going to get, I'm going Say it's equal to the set, it's the set of all vectors x, that are members of R n, and I'm gonna double down So the null space of A, the null space of A, is equal to, or I could Independence of these vectors, to the null space of A. Now what I want to do, I said I want to relate the linear ![]() And remember, each of these,Īre going to have m terms, or I should say, m components in them. Well this is going to be V one for that column, V one for that column, V two for this column, all the way, we're gonna have n columns, so you're gonna have V That that's a matrix, we could rewrite it as, so let me do it the same way, so, draw my little brackets there, we can write it, just express it, in terms of its column vectors, cos we could just say And so we could rewrite A, weĬould rewrite the matrix A, the m by n matrix A, I'm bolding it to show This would be V two, V two, and you would have n of these,īecause we have n columns, and so this one right And so, let me do it this way, so you can view this one right over here, we could write that as V one, V one, this next one over here, There's n columns here, and we could view each of thoseĪs an m-dimensional vector. So what, first of all what I am talking about as column vectors? Well as you can see Or linear dependence, of the column vectors ofĪ, to the null space of A. And what I want to do in this video, is relate the linear independence, The matrix A over here, and A has m rows and n columns, so we could call this an m by n matrix.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |