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
4) Determinants
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
8) Isometries
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:
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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)
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Additional topics such as Jordan form of matrices, quotient vector spaces, …
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Good exercises (preferred with solutions somewhere)
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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.
Best Answer
Advanced Linear Algebra by Roman is a nice book: http://www.springer.com/gp/book/9780387728285.
If you're looking for connections between linear algebra and geometry, then I'd recommend just immediately studying differential geometry. A good place to start here is An Introduction to Manifolds by Tu: http://www.springer.com/gp/book/9781441973993.
The study of infinite dimensional vector spaces is called functional analysis and a good place to start here is Introductory Functional Analysis with Applications by Kreyszig: http://au.wiley.com/WileyCDA/WileyTitle/productCd-0471504599.html.