[ajkjk]

You have unfortunately written off everything I said by assuming the most naive understanding of it.

The general theme is that I am interested in the metaphysical concept of vectors, not the thing that human mathematicians have labeled vectors. The universe doesn't care if you write ax+by or x^a y^b, hence addition vs multiplication is just a choice of coordinate system. And matrices and functions are vector spaces sure, but out in the world, when they show up in modeling things, they are local linearizations of curved things. Every linear algebra is (inevitably) a local point in a nonlinear one, as far as I can tell. Not in a formal sense, but in the sense that when you go out into the world and find them, it turns out to be the case.

The general theme is: I don't want to spend my life mastering the rigor of these simplistic models so that I can do it intuitively (in Tao's sense); I want to use them to learn intuition of the things that they are simplistic models of, and then master that.

[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)).