I was pondering what it means to understand math… consider Linear Algebra, a relatively simple abstract theory with many applications.
I studied linear algebra and got a decent grade. I know how to use most of the (elementary) theory to solve problems. But I don’t feel I understand linear algebra.
For example take determinants. To me this is just a peculiar rule to assign a number to a matrix that has many useful uses. Of course this grants the determinant a special status among other inconsequential rules that could be devised to assign numbers to matrices. But I don’t know what a determinant is or why it is the way it is. A determinant to me is a weird thing. I can’t visualize it. And looking at the definition I haven’t got the faintest clue what permutations and whatnot has to do with anything.
So would a mathematician be, compared to me, just more proficient in memorizing and applying these definitions and theorems or would a determinant be a familiar entity to him? Would he see why a determinant HAS to be zero for a singular matrix? (Well, not just because it follows from the definition of course).
When I look at a real integral… even though I might have trouble calculating a complicated integral analytically, I have no problem visualizing what it represents. I “get” integrals and derivatives. It helps that I can sketch them in a graph. I don’t “get” determinants, eigenvectors, kernels, etc.
They are more abstract concepts, so that would presuppose more effort. So am I just unable to elevate my understanding to a higher level or would a mathematician share my feelings?