[Tainnor]

> The general theme is that I am interested in the metaphysical concept of vectors, not the thing that human mathematicians have labeled vectors.

That's fine but then you shouldn't be surprised that you can't read higher level mathematics textbooks, because those are not about metaphysics.

The rest of what you wrote is... just not true. Matrices are used in plenty of areas where they are not mere approximations (e.g. cryptography), and spaces of functions remain vector spaces even when the functions themselves are not linear (because there's a difference between requiring that f(x+y)=f(x)+f(y) and that (f+g)(x) = f(x)+g(x)).