A short interview with Gilbert Strang about teaching linear algebra, centered on the "four fundamental subspaces." Strang defines a matrix as a rectangle of numbers, describes vectors as columns of numbers (abstract, possibly high-dimensional), and builds intuition for the column space and row space (all linear combinations of columns/rows). He then introduces the two perpendicular spaces (null spaces) that complement the column and row spaces, presenting the four subspaces as a unified, visualizable picture of a matrix.
Chapters
So let's talk about linear algebra a little bit because 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 spaces?
Sure I can, yeah. So the first one to understand is. So the matrix, maybe I should say the matrix is.
What is a matrix?
What's a matrix? Well, so we have a, 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 got to be connections there, but they're not simple. The, 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 the. How it's very commonly used, perhaps. Yeah. You think this arbitrary space, multidimensional space.
Right away I'm in high dimensions and in the 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 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, 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 and. Yeah, how do you think about the things like a 10 dimensional vector that we can't really visualize.
Yeah.
Do you? Do you? And, and yet math reveals some beauty. Oh, underlying.
Yeah.
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, is that it goes on to 10 dimensions with no problem. I mean that if you're just seeing what happens if you add two vectors in 3D, 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. 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. 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, 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, everybody likes that railroad chalk. And me too. I wouldn' Anything else now. Yeah. And, 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 the, in a beautiful picture of a matrix. Yeah, yeah. It's sort of a fundamental, it's not a difficult idea. Comes, comes pretty early in 1806. And it basically.
