Linear Algebra book (useful for advanced algebra courses) First of all, I know that there are many posts on this topic, but my question, as you will see, has important differences.
I am in the last year of my undergraduate studies, and I have been taught introductory courses of Group Theory and Ring Theory. Now, I want to attend Galois Theory, but I see that I have no good knowledge of Linear Algebra. So, I thought that this is a good chance to refresh and complete my knowledge. And I look for a good book with many exercises to solve.
In other words, I'm looking for a good Linear Algebra book, which will have all the necessary chapters with exercises and will be useful for Galois Theory, Algebraic Geometry, Representation Theory, Coding Theory and other more advanced algebra courses.
What are your recommendations?
PS: I didn't like Gilbert Strang's book "Linear Algebra and Its Applications".
PPS: What's your opinion for the books: 1) Linear Algebra, Serge Lang, 2) Introduction to Linear Algebra, Serge Lang and more generally your opinion about Lang's books, 3) Linear Algebra: Step by Step, Kuldeep Singh?
PPS: I apologize for my English!
Thank you in advance!
 A: I think the best book for you will be Linear Algebra done Right by Sheldon Axler. The reasons being
1.) It takes an abstract but intuitive approach, is heavy on proofs.
2.) Exercises are not too numerous, about 20-25 in each chapter and are not unreasonably hard; and cover most of the techniques used in the chapter.
3.) Since you will be doing a course on galois theory, you will find the discussions in the later chapter on minimal polynomial etc illuminating.
4.) Finally, I was in a similar state 6 months back and had to pick up linear algebra properly before course on galois theory. It took me about 10 days to finish it. It is not dense but is still quit thin. And the writing is superlative when compared to other books on the same topic.
Rest other books that you have mentioned have one or two of the flaws that I mentioned that Axler doesn't have. GO for it.    
A: I think it better to study both algebraic and geometric intuition of linear algebra, it helps you to understand how theorems work, in this way you can apply Linear algebra technique in other fields. 
For geometric intuition: I recommend "Essence of linear algebra"  series which aimed at animating the geometric intuitions.
For Algebraic intuition: you can use Stephen Boyd and Lieven Vandenberghe book 
"Introduction to Applied Linear Algebra" , It is used as the textbook for the course EE103 (Stanford) and EE133A(UCLA).
indeed, you can use Sheldon Axler book " Linear Algebra Done Right" for more algebraic intuition.
A: If you want a self-study book on linear algebra then go for 'Linear Algebra Step by Step' as it has complete solutions to all the problems online.
