Linear Algebra Miniatures: Day 3
This semester I am teaching a class called “Linear Algebra and Differential Equations”, so I thought I’d spend a couple minutes at the beginning of classes talking about linear algebra. (The full backstory can be found here)
For those of you coming from the linear algebra sequence, we’re going to focus primarily on vector spaces instead of the full force of modules; Most things I’m going to say are also true at least for free modules, but definitely not everything and I’ll do my best to highlight where those differences arise.
So at some point in your life, you’ve probably heard about function composition. Probably for the first time in precalc and then you got a lot of practice with it in calc from the chain rule, u-substitutions, change of variables, and so on. In any case, I’ll remind you how it works: we have two functions $f$ and $g$ and we want to make a third function $f\circ g$:
$$ (f\circ g)(x) = f(g(x)) $$
so you take the input, put it into $g$, and whatever it gives out you put that immediately into $f$, and whatever comes out of $f$ is the number you want.
Not often covered in a precalc class is the following issue:
$$ \begin{align*} f(x) &= 3\log(x) \ g(x) &= \begin{bmatrix} x \ 2x-1 \end{bmatrix} \end{align*} $$
So I’ve defined two perfectly good functions, but if I try to take $f\circ g$, we have to take the logarithm of a vector— of course that doesn’t make any sense. We say that $f$ and $g$ aren’t compatible. Why not? Well, the key here is that the sorts of objects that $f$ takes in need to be the same types of objects that $g$ spits out.
But now we know— or at least I’ve said it enough times that maybe you’re starting to believe me— that there is this relationship between matrices and linear maps. So if $F$ and $G$ are not just functions, but linear maps, you can’t stop me from taking the matrix associated to $F$ and the matrix associated to $G$ and the matrix associated to $F\circ G$.
But of course the three maps weren’t just random, the last one has a very strong relationship to the first two. We’d like to be able to say the same for the matrices. In other words, we want to fill in the “missing” arrow on the right-hand side.
Of course, we can draw that arrow, which is given precisely by matrix multiplication: $C=AB$. So if I were in the mood to make grandiose statements, I could say “Therefore, in order to understand matrix multiplication, we must first understand the relationship between matrices and linear maps.”
But of course that’s completely false…
I mean, it has to be, right? Because we did talk about matrix multiplication in lecture yesterday and we didn’t talk about linear maps. And really, that’s the whole point of matrix multiplication: we don’t want to go the long way around the square, and matrix multiplication says we don’t have to. We can stay on the right side of the picture without ever even realizing there is a left side.
But this does, I think, explain why matrix multiplication is so weird and counterintuitive; it’s not really an operation on matrices. It’s an operation on linear maps that we forced to work for matrices. And it ends up being a pretty heavy labor-saving device, so we’re going to go through some effort to really understand it.