Comment by throwawaymath

Yes, I think you've captured the problem pretty nicely. This is the problem I have with doing something like defining vector spaces on arbitrary fields F (not even R or C!) before you even talk about linear equations. The theory of linear algebra is rich and useful, but it's hard to motivate it that way. I don't mean "motivate" in the sense of, "Why should I care about this, and when will I use it?", because you don't always need that kind of motivation to do math. I mean motivation in the sense of, "Why did people come up with this theory? What led them to these definitions, and why are these definitions the right ones?"

I have seen the hierarchy of math presented like this:

1. In high school, you're taught how to compute arc length. You need to do nothing but calculation, and it doesn't matter why it works.

2. In undergrad, you're taught to prove the formula for computing arc length. Given the theorem statement and the requisite axioms, you can show that the formula works.

3. In grad school, you're taught to derive the formula for arc length from first principles. Given a set of axioms, you come up with the theorem and prove it from scratch.

4. In research, you're not taught anything. Instead you solve the questions, "How should I define arc length? Why does my definition of arc length matter, and where is it useful? What can I prove with it?"

Contrary to (quite a bit of) popular opinion, I would hold that a truly rich understanding of the theory can only happen in the context of the trigger for the theory. You don't need to understand the context of systems of linear equations, or why linearity is a useful concept, to understand vector spaces. But it's a lot easier if you have that context. Likewise you don't need to use Gaussian elimination to solve a lot of questions which ask you to prove something about a vector space or a linear map. But if you have that context, you can use it in your proofs instead of miring in towering heights of complexity.