Very good linear algebra book. I plan to self-study linear algebra this summer. I am sorta already familiar with vectors, vector spaces and subspaces and I am really interested in everything about matrices (diagonalization, ...), linear maps and their matrix representation and eigenvectors and eigenvalues. I am looking for a book that handles every of the aforementioned topics in details. I also want to build a solid basis of the mathematical way of thinking to get ready to an exciting abstract algebra next semester, so my main aim is to work on proofs for somehow hard problems. I got Lang's "Intro. to Linear Algebra" and it is too easy, superficial.
Can you advise me a good book for all of the above? Please take into consideration that it is for self-study, so that it' gotta work on its own. Thanks.

old, but good: Linear Algebra and its Applications by Gilbert Strang, see http://www.amazon.com/Linear-Algebra-Its-Applications-Edition/dp/0155510053/ref=sr_1_4?ie=UTF8&qid=1400721854&sr=8-4&keywords=strang+algebra
