You've taken the usual approach of explaining matrix operations as an unmotivated "this is what we do" and then tried to layer linear transformations as a weird abstract thing which somehow relates. Most people do this, and it makes sense for people who don't want to actually understand.
If you want people to actually understand, the relationship goes the other way. Linear transformations are the thing we really care about, and matrices happen to be the most convenient way to write them.
More specifically if T is a linear transformation from V to W, with bases (v_1, ..., v_n) and (w_1, ..., w_n), then then we can write down M = (T(v_1) T(v_2) ... T(v_n)). So far that's just a rectangular array of numbers, uniquely defined for the linear transformation. Now if we have a vector v = a_1 v_1 + a_2 v_2 + ... + a_n v_n then just tracing through linearity, we get that T(v) is Mv with the regular matrix multiplication. Furthermore we can now prove that any rectangular matrix defines a unique linear function with that rule.
The result? When you have 2 vector spaces with a basis for each, each linear function gives a unique matrix, and each matrix gives a unique linear function. Matrices are just convenient ways of writing down linear transformations.
OK, now if S is a linear transformation from W to U, and we have a basis for U, then what about the function S o T defined by S o T(v) = S(T(v))? If you work it out, it is a linear function, and its matrix is the usual matrix multiplication. So matrix multiplication is the same as function composition for linear functions.
Here is something fun. You know the associative law? That magically just works after a ton of calculations for matrices? Well, here is why it works. For any three linear transformations R, S, and T which can be composed, and any vector v, by definition:
((R o S) o T)(v) = (R o S)(T(v)) = R(S(T(v))) (R o (S o T))(v) = R((S o T)(v)) = R(S(T(v)))
Which means that functions under composition are associative. Since matrix multiplication is just a way of writing function composition, it must also be associative. QED!
As you move through things like change of basis matrices, it is all a lot easier to understand starting from the point of view of linear functions as being fundamental, and matrices just being a way to represent it with a given pair of bases.
If your goal is real understanding, the approach that I stated is the right one. If you've learned it the other way and wish to actually understand it, you'll still have to learn the approach that I stated.
Your goal is explicitly not real understanding. And most textbooks do it in the order that you did. (Partly because professors think that most students aren't going to try to understand, so there is no point in giving explanation.) So you probably made the right choice.
But I would still like to see even a passing mention that matrices of just a representation of a linear function given two bases, and matrix multiplication is function composition. That could set a lightbulb on for someone struggling through a problem.
https://news.ycombinator.com/item?id=14480559 https://news.ycombinator.com/item?id=14497867 https://news.ycombinator.com/item?id=14549007
As for the rest, you might be right or not; but I am not competent to reform world's math education. If you can write better linear algebra guides than the standard textbooks, I'm all for it. Please inform me if you do that, I'd love to read that. (BTW It's not sarcasm - I'd really love to see linear algebra explained better than in standard textbooks. So far all fancy tutorials turned to be good only as entertainment).
Let me offer an abstract example. Consider the polynomials of degree at most 2. There is an obvious basis, namely 1, x, and x^2. A polynomial like x^5 - 3x + 2 can now easily be written in coordinates as (2, -3, 1).
However we have many other coordinate systems that might be convenient. For example suppose that we're sampling data, and can measure p(0), p(1) and p(2). How do we find what polynomial that is? Here is an easy way. We can easily find the new coordinates for our basis vectors: 1 -> (1, 1, 1), x -> (0, 1, 2), x^2 -> (0, 1, 4). That means that we can write down the matrix representing the identity transform (nothing happened), going from the basis we have, to the new coordinate system:
( 1 1 1)
( 1 2 3)
( 1 2 4)
( 2 -1 0)
(-3 2 -1)
( 1 -2 1)
There are a lot of problems where linear algebra comes up that you can think through more clearly if you think about things this way (complete with the role of the basis!) than if it isn't fully digested.
As for a better book, well, I already recommended Down With Determinants! :-)
Maybe I'm hopelessly lost, but, for one, the inverse of
( 1 1 1)
( 1 2 3)
( 1 2 4)
( 2 -1 0)
(-3 2 -1)
( 1 -2 1)
Anyway, who is missing the role of the basis? It's in practically every textbook, including the one I used.
I agree with the maths that you've written, but it's the same thing that the textbook explains, practically in the same way. The road to understanding is not the same for all people, I guess...
As I commented in email, I did it by hand while very tired and made multiple mistakes.
There are now videos! http://www.linear.axler.net/LADRvideos.html