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.
 A: Jim Hefferon at Saint Michael's College, has a pretty well known linear algebra textbook that he provides for free: Linear Algebra.
A: Linear Algebra and Its Applications, 4e by David C. Lay
This is the #1 rated Linear Algebra book on Amazon. It should be good! I'm using it for a class next semester here at UW, which is ranked #9 in the country for Mathematics.
A: I can't believe nobody has mentioned Peter Lax's Linear Algebra and Its Applications. 
A: When I learned linear algebra for the first time, I read through Friedberg, Insel, and Spence. It is slightly more modern than Hoffman/Kunze, is fully rigorous, and has a bunch of useful exercises to work through.
A: I can recommend you two books. One already mentioned by @JPi and it is Linear Algebra and Its Applications by G. Strang: http://www.amazon.com/dp/0155510053/?tag=stackoverfl08-20
Another book that I like very much is Fundamentals of Matrix Computations by D. S. Watkins: http://www.amazon.com/Fundamentals-Matrix-Computations-David-Watkins/dp/0470528338 You can find there fundamental algorithms that are used in the field of matrix computations.
A: If you are new to linear algebra ,then you should use  "Introduction to Linear Algebra" by Gilbert Strang.In case you posses some knowledge of LA then you can use " Matrix Theory and Linear Algebra" by I.N. Herstein .There are many books on pure linear algebra and computational linear algebra,you can choose as per your requirement and interest.
The two books i have recommended they can serve as foundation for both.
A: Carl Meyer's Matrix Analysis and Applied Linear Algebra is my favourite. While many of the above books are good, Meyer's book has a great focus on how you actually find the various objects - this is in stark contrast to Hoffman-Kunze who are content with existence proofs. 
A: A great book freely available online is Linear Algebra Done Wrong by Sergei Treil. It covers all the topics you listed and culminates in a discussion of spectral theory, which can be considered a generalized treatment of diagonalization.
Don't be put off by the book's title. It's a play on the popular Linear Algebra Done Right, by Sheldon Axler. Axler's book is also very good, and you might want to check it out.
The classic proof-based linear algebra text is the one by Hoffman and Kunze. I find the two books I listed above easier to read, but you might also consider it. In any case, it is a good reference.
I hope this helps. Please comment if you have any questions.
A: Linear Algebra by Fraleigh is a good book.
A: S. Winitzki, Linear Algebra via Exterior Products (free book, coordinate-free approach)
A: Linear Algebra by Hoffman-Kunze. Might be a little too deep, but I believe you'll do fine with it.
A: 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
