This may sound like a philosophical question, but it’s intended as a very practical question.
Broadly, I have two ways of doing math:
blindly following definitions and shifting equations around.
having a deep understanding of the meaning of terms and why theorems hold, allowing me to see immediately why a conjecture must hold, or what the solution to a problem will look like. I may not even need to do any formal derivations, and if I do, they flow immedaitely from my intuitive understanding.
With respect to linear algebra (and abstract algebra, though this question focuses more on linear algebra), I am to a large extent in (1), and I want to get to (2).
Mainly due to watching a 3blue1brown series, I have an intuitive understanding of
- what an eigenvalue and eigenvector is geometrically
- that a matrix represents geometrically, a linear operation on e.g. a Euclidean vector space
- other basic stuff
But when it comes to other concepts, whenever I use them I am really just shifting equations around mindlessly:
- matrices can be decomposed into a diagonal matrix and two other matrices that are inverses of eachother
- what a quadratic form represents geometrically
- that symmetric positive definite matrices have symmetric square roots, but positive definite matrices don’t necessarily have symmetric square roots.
- that symmetric matrices have real eigenvalues The sum of eigenvalues of a matrix are equal to its trace, and the product of eigenvalues equal to the determinant.
- etc etc
Alot of these things I can prove by mindlessly sequencing equations like a computer-based theorem prover, but I cannot immediately see why they are or are not true.
What can I do >>practically<< to gain this type of intuition quickly about lots of these linear algebra questions?
I feel like just doing more theorems, and solving more problems, won’t help. Are there good books that treat these things intuitively? Or a collection of methods to use?
(Note that I am entirely self-taught in this and have no formal math degree).