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.

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 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.