I'm a mathematics undergrad student who finished his first university year succesfully. For this post, it should be interesting to note that I already took a course in group theory.
I also had a course in linear algebra where we treated the following topics:
1) Vector spaces; basis; span; linear dependency; direct sum
2) Linear transformations, matrices, rank
3) Linear varieties, system of linear equations
5) Eigenvalues and eigenvectors, diagonalisation, triangalisation, cayley-hamilton,
6) Euclidean spaces: inner products, norm, hermitian transformations, orthogonal basis, Gramm-Schmidt, orthogonal transformations
7) Prehilbert spaces
9) Bilinear forms and quadrics in $2$ and $3$ dimensions
Now, I want to revise these topics, but it would be too boring to reread these books again, so now I look for a book that has all these topics, but also some additional requirements:
Focus on intuition (e.g. connections between linear algebra and geometry). This is a hard requirement for me, since there was no attention given to this in the course I took)
Additional topics such as Jordan form of matrices, quotient vector spaces, ...
Good exercises (preferred with solutions somewhere)
The book must be rigorous
In the course I had, we mainly discussed finite-dimensional vector spaces, but I am also interested in a theory of infinite dimensional vector spaces.
Can someone hint me towards a good book that would suit me? If I have to add any information, please leave a comment and I will edit my post.
Thank you for your time.