I went through EE and CS. EE we started using matrices exactly how you describe it: here’s a system of linear equations, here’s how you write them in matrix form, here’s how you invert them to solve the original system. Turn the crank, answer pops out. I had my trusty HP49G, and I could solve linear systems all day.

Then in CS I took a computer graphics course and it was rotation and translation matrices all day every day.

Then there was a digital communications course where we touched on orthogonal basis functions, and some matrix voodoo related to that and how to get orthogonal vectors out of the mess.

And then finally I took the required CS linear algebra course offered by the math department, where we started from scratch. Here’s a vector (psh, I know vectors!), here’s a vector space (hmmm this is new), and building the rest of it up from there. I really wish that had come earlier on, but I was very very happy to finally have a bit of a theoretical understanding of how these tools I’d been using actually worked.