- So let's talk about linear algebra a little bit, 'cause it is such a, it's both a powerful and a beautiful subfield of mathematics. So what's your favorite specific topic in linear algebra, or even math in general, to give a lecture on, to convey, to tell a story, to teach students?

- Okay, well, on the teaching side, so it's not deep mathematics at all, but I'm kind of proud of the idea of the four subspaces, the four fundamental subspaces, which are, of course, known before, long before my name for them, but-- - Can you go through them? Can you go through the four subspaces?

- Sure I can, yeah. So the first one to understand is, so the matrix, maybe I should say the matrix-- - What is a matrix? - What's a matrix? Well, so we have like a rectangle of numbers. So it's got N columns, got a bunch of columns, and also got an M rows, let's say, and the relation between, so of course, the columns and the rows, it's the same numbers.

So there's gotta be connections there, but they're not simple. The columns might be longer than the rows, and they're all different, the numbers are mixed up. First space to think about is, take the columns, so those are vectors, those are points in N dimensions. - What's a vector? - So a physicist would imagine a vector, or might imagine a vector as a arrow in space, or the point it ends at in space.

For me, it's a column of numbers. - You often think of, this is very interesting in terms of linear algebra, in terms of a vector. You think a little bit more abstract than how it's very commonly used, perhaps. You think this arbitrary multidimensional space-- - Right away, I'm in high dimensions.

- Dreamland. - Yeah, that's right, in the lecture, I try to, so if you think of two vectors in 10 dimensions, I'll do this in class, and I'll readily admit that I have no good image in my mind of a vector, of a arrow in 10 dimensional space, but whatever.

You can add one bunch of 10 numbers to another bunch of 10 numbers, so you can add a vector to a vector, and you can multiply a vector by three, and that's, if you know how to do those, you've got linear algebra. - You know, 10 dimensions. - Yeah.

- You know, there's this beautiful thing about math, if we look at string theory, and all these theories which are really fundamentally derived through math, but are very difficult to visualize. How do you think about the things, like a 10 dimensional vector, that we can't really visualize? - Yeah.

- Do you, and yet, math reveals some beauty-- - Oh, great beauty. - Underlying our world in that weird thing we can't visualize. How do you think about that difference? - Well, probably, I'm not a very geometric person, so I'm probably thinking in three dimensions, and the beauty of linear algebra is that it goes on to 10 dimensions with no problem.

I mean, if you're just seeing what happens if you add two vectors in 3D, yeah, then you can add them in 10D. You're just adding the 10 components. So I can't say that I have a picture, but yet I try to push the class to think of a flat surface in 10 dimensions.

So a plane in 10 dimensions, and so that's one of the spaces. Take all the columns of the matrix, take all their combinations, so much of this column, so much of this one, then if you put all those together, you get some kind of a flat surface that I call a vector space, space of vectors, and my imagination is just seeing like a piece of paper in 3D.

But anyway, so that's one of the spaces, that's space number one, the column space of the matrix. And then there's the row space, which is, as I said, different, but came from the same numbers. So we got the column space, all combinations of the columns, and then we got the row space, all combinations of the rows.

So those words are easy for me to say, and I can't really draw them on a blackboard, but I try with my thick chalk. Everybody likes that railroad chalk, and me too. I wouldn't use anything else now. And then the other two spaces are perpendicular to those. So like if you have a plane in 3D, just a plane is just a flat surface in 3D, then perpendicular to that plane would be a line.

So that would be the null space. So we've got two, we've got a column space, a row space, and there are two perpendicular spaces. So those four fit together in a beautiful picture of a matrix. Yeah, yeah, it's sort of a fundamental, it's not a difficult idea. It comes pretty early in 1806, and it's basic.

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