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OK, this is linear algebra lecture nine. And this is a key lecture, this is where we get these ideas of linear independence, when a bunch of vectors are independent -- or dependent, that's the opposite. The space they span. A basis for a subspace or a basis for a vector space, that's a central idea. And then the dimension of that subspace. So this is the day that those words get assigned clear meanings. And emphasize that we talk about a bunch of vectors being independent. Wouldn't talk about a matrix being independent. A bunch of vectors being independent. A bunch of vectors spanning a space. A bunch of vectors being a basis. And the dimension is some number. OK, so what are the definitions? Can I begin with a fact, a highly important fact, that, I didn't call directly attention to earlier. Suppose I have a matrix and I look at Ax equals zero. Suppose the matrix has a lot of columns, so that n is bigger than m. So I'm looking at n equations -- I mean,...

OK, here's linear algebra lecture seven. I've been talking about vector spaces and specially the null space of a matrix and the column space of a matrix. What's in those spaces. Now I want to actually describe them. How do you describe all the vectors that are in those spaces? How do you compute these things? So this is the, turning the idea, the definition, into an algorithm. What's the algorithm for solving A x =0? So that's the null space that I'm interested in. So can I take a particular matrix A and describe the natural algorithm, and I'll execute it for that matrix -- here we go. So let me take the matrix as an example. So we're definitely talking rectangular matrices in this chapter. So I'll make, I'll have four columns. And three rows. Two four six eight and three six eight ten. OK. If I just look at those columns, and rows, well, I notice right away that column two is a multiple of column one. It's in the same direction...

Okay, this is linear algebra, lecture four. And, the first thing I have to do is something that was on the list for last time, but here it is now. What's the inverse of a product? If I multiply two matrices together and I know their inverses, how do I get the inverse of A times B? So I know what inverses mean for a single matrix A and for a matrix B. What matrix do I multiply by to get the identity if I have A B? Okay, that'll be simple but so basic. Then I'm going to use that to -- I will have a product of matrices and the product that we'll meet will be these elimination matrices and the net result of today's lectures is the big formula for elimination, so the net result of today's lecture is this great way to look at Gaussian elimination. We know that we get from A to U by elimination. We know the steps -- but now we get the right way to look at it, A equals L U. So that's the high point for today. Okay. Can I take the easy part, the first s...

OK. This is linear algebra lecture eleven. And at the end of lecture ten, I was talking about some vector spaces, but they're -- the things in those vector spaces were not what we usually call vectors. Nevertheless, you could add them and you could multiply by numbers, so we can call them vectors. I think the example I was working with they were matrices. So the -- so we had like a matrix space, the space of all three by three matrices. And I'd like to just pick up on that, because -- we've been so specific about n dimensional space here, and you really want to see that the same ideas work as long as you can add and multiply by scalars. So these new, new vector spaces, the example I took was the space M of all three by three matrices. OK. I can add them, I can multiply by scalars. I can multiply two of them together, but I don't do that. That's not part of the vector space picture. The vector space part is just adding the matrices and multiplying by nu...