[nostalgebraist]

I took linear algebra as an undergrad and then took a slightly fancier version in my first year of grad school, and I understood all the “matrices <==> linear transformations” stuff, but I never really felt comfortable interpreting the actual entries of a matrix until my second year of grad school, when I learned the rule

the matrix-vector product A*v is a linear combination of the columns of A, with the coefficients given by the entries of v

I learned this from the excellent book Numerical Linear Algebra by Trefethen and Bau, and I don’t think I’ve ever heard it mentioned by anyone else outside of that book.  Yet it’s been invaluable to me, and not just for numerics.  Did I just miss out, or is this simple fact not disseminated widely enough?

[jadagul]

The difficult thing about teaching linear algebra (he says, procrastinating from writing the last week of notes for the linear algebra class he is teaching) is that the entire subject is, like, four actual facts, each of which is repeated twenty times in slightly different language.

And you have a great example! We could talk about:

1. A linear transformation, as a function with certain properties.

2. Matrix mutiplication

3. A system of linear equations

4. A collection of dot products with the row vectors

5. A linear combination of column vectors

6. A hyperplane in some higher-dimensional space

7. A semi-rigid geometric transformation of some space.

8. A function determined entirely by what it does to some basis.

And those are all the same thing. I think typically students coming out of a (first) linear algebra class understand and have internalized a couple of those; can cite a couple others; and are completely oblivious to the rest. (Any may not have heard of some, because it’s hard to cover all eight; I know that my discussion of the geometric properties has been somewhat perfunctory.

But for any given person, some of these perspectives will make much more sense than others; and if your class doesn’t get you to the ones that work for you, you won’t understand nearly as much as if it does.

(The goal, of course, is to understand all of the perspectives, and to switch among them fluently, but that’s hard and definitely not happening in a first course. So you have to pick your focuses. The reason I was so unhappy with my college’s choice of textbook is that its focus is exacty the opposite of what I would like).

So, for instance, you say that the matrix product is a linear combination of the columns of the matrix, with coefficients given by the input vector. And you say that, and I think for a few seconds and say “huh, I guess that’s true.” But that’s not how I think about it; I think about it as a function that sends each standard basis element to the corresponding column vector.

Except those are literally the exact same thing. You write your input as a linear combination of your standard basis vectors, and then your function preserves linear combinations, and sends each basis vector to the corresponding column—so you get a linear combination of the column vectors.

And I think the thing I just said is pretty common to mention. It’s certainly necessary for doing any sort of change-of-basis stuff. But if it made more sense to you in different language, that’s 100% unsurprising.