I am looking for a linear algebra book which is as abstract as possible. But not an abstract algebra book. Something that is what Rudin's is for (beginning) analysis, that is, terse, rigorous and with beautiful exercises.
For example: Intro to LA, by Curtis, and LA by Hoffman-Kunze. Opinions?
(I know there are already very similar questions around, but I felt they are not quite the same.)
Best Answer
I am not sure what you are really looking for ("as abstract as possible" would suggest a Bourbakist approach, but this is not likely to involve many "beautiful exercises", though in part this depends on what you consider beautiful and what you consider and exercise). But since nobody else did yet, I'd like to mention Roger Godement's Cours d'Algèbre (1963), which would meet some of your criteria (and has been translated in English, too). It is not restricted to linear algebra, but those willing to admit some basic stuff about logic, set theory, groups, rings, and complex numbers can start reading at section 10 (modules and vector spaces), after which it is pretty much all stuff relevant to Linear Algebra until the final section 36 (Hermitian forms).
In the Bourbakist tradition vector spaces are of course introduced as a special case of modules over a ring. However the pursuit of generality is done only where it is painless, in the sense that it assumes the level of generality natural for the theorems being considered. This means for instance that in section 19 (the notion of dimension) the base ring is assumed to be a skew field (a.k.a. division ring; just "corps" in French where this term does not imply commutativity) whereas in the previous section (finiteness theorems) it was only assumed to be Noetherian or principal, depending on the exact results stated. In the lead-up to determinants (sections 21-24) the ring or field will be assumed commutative, and sections 34-36 where eigenvalue problems are discussed assume a true (commutative) field. One of the really nice things that I found in this approach is that not needlessly assuming commutativity forces some notational habits that I have found are useful even when working over rings that are commutative (for instance writing scalars at the opposite side of vectors than linear maps).
For the record, the book contains about 165 pages of exercises at the end, and though I cannot vouch for the beauty of all of them, there must be some nice ones to found there.