"The eigenvectors are the axes of the transformation" = mind blown. After several engineering courses, studying eigenvectors in an advanced math class and still no one could put it this simple. This guy is amazing.
Even then, it was a question I had to ask my brilliant, constantly-pissed looking young professor. "Hey, uh, the Jacobian... what does the determinant mean, uh, geometrically?". He looked at me like a slug, before explaining it was the measure of the newly mapped unit square. Fireworks went off in my head. Two linear algebra classes before only ever explained it by its algorithm or its usefulness (e.g., ∃ A^(-1) for A \in R^{n,n} iff det(A) != 0)
Side note: that professor had the most effective teaching style for pure math I've ever seen. Besides lectures that expanded on the contents of Rudin and interesting problem sets, he gave us a list of a hundred theorems, propositions, and exercises. Told us the final exam would be six problems, four of which would come from that list, another of which would be a clever new one, the last something truly hard.
Never learned analysis better than when sitting down and working through (not memorizing) each of those proofs and theorems for possible later recapitulation.
FWIW, that book has two explanations: the first on p 296 is a lot like "that's how it works out" and the second on p 320 is geometric.
> why does the determinant work to solve linear equations (i.e. Cramer's rule)?
Does the explanation on p 331 of joshua.smcvt.edu/linearalgebra/book.pdf help? (It uses the geometric understanding of the determinant.)
Recently, learning to do Principal Components Analysis to solve a handwriting recognition problem is what finally shed an enchanted light on linear algebra. The way that a simple matrix of data samples is transformed into an ordered set of principle components (eigenvectors, ordered by eigenvalue) is.. "unreasonably effective" (as they say). The principal components are your signal, and the rest (with eigenvalues ~0) are the noise. the handwriting recognition, btw, works fantastic for my simple application. no need for non-linear kernels and whatnot.
Even better now, in this hacking life after pure math in college and grad school, is that I can build intuition now not just by proofs and exercises, but also efficient, coded implementation. Gives a different feel for the tools and concepts.