To me, one of the interesting things about how mathematical results have been discovered historically is that they have often been accidental (or intended for something else).

eg. eigenvalues were originally looked at for use with quadratic forms, but people later found that its interesting properties (orthogonality, symmetry, etc) were useful for lots of other things.

There is often also a large gap in time between when a mathematician playing around with a problem discovers an interesting result and when it is actually applied to something useful.

eg. Euler's law regarding complex exponentials was developed around 1740, but it wasn't until around 1807 when Fourier used it in harmonic analysis and 1897 when Steinmetz started applying it to electrical engineering.

But you wouldn't know that from reading a textbook, which is (understandably) arranged in such a way that omits historical context and why people bothered to study it in the first place. Most linear algebra books are classic examples of how to introduce abstract topics without any context.

Not me. Well, I'm a post-grad and my Ph.D. is quite mathsy. That doesn't make me a mathematician (I still believe that the title should be earned and I definitely have not yet earned it), but at least I've done some mathematics.

Anyway, I always find it much easier to move from the concrete to the abstract. But once you've mastered an abstraction, it does seem to become "concrete" in your brain and then you can build on it. Thus, with a good background in basic linear algebra, you can move on to tensor analysis, but good luck if you want to jump into tensors straight away.

My abstract algebra book from college introduced the axioms and used them right away. But new concepts were introduced independently and afterwards related to the axioms.