Seeing as Axler is very reluctant to talk about determinants and generally avoids computations and playing around with algebra, I'd like to get a book that will serve as a companion to Axler's unorthodox approach.
I'm not terribly interested in real-world applications. I don't much mind them, but the maths is what matters most to me.
All things considered, the book should certainly be mathematically rigorous without being particularly advanced – a first course with some minor prior exposure is assumed here. However, the reader has a strong foundation in proofs. The book should serve as a counterpart to Axler's approach; the book should, among other things, make up for the topics Axler lacks in.
I'd love to hear your suggestions, especially from people who have experience with Axler. The target audience is a pure maths major. Again, applications aren't necessarily a deal-breaker, but I'm not interested in lots of trivial exercises that exist only to show the real-world value of linear algebra; that, I do consider a deal-breaker.
Best Answer
I think Linear Algebra by Friedberg, Insel, and Spence is a careful, clear, and very standard/orthodox treatment of Linear Algebra.