His description of the determinant, too. I hadn't heard that explanation until a second semester of real analysis, when learning the proof of the Inverse Function Theorem (an amazing thing to study, by the way, connects many the dots between linear algebra and calculus).

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.