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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:

  • 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.

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  • $\begingroup$ I find that the more abstract you go, the more difficult "intuition" is, especially when you move away from finite dimensional vector spaces. Anyway, if you like, Linear Algebra Done Right by Sheldon Axler is a good start to the abstraction. $\endgroup$ – Sean Roberson Aug 22 '17 at 20:28
  • $\begingroup$ Check out Babai and Frankl [link] matthewkahle.org/download/file/fid/499 ... I would also suggest you add some motivation as to why you want to learn linear algebra (grad school aspirations, industry, etc) $\endgroup$ – mm8511 Aug 22 '17 at 20:30
  • $\begingroup$ Looking at field theory (abstract algebra, algebraic number theory) might be a good idea, as it uses a lot of groups and linear algebra in arbitrary fields. Another topic is functional analysis (linear operators in Banach and Hilbert spaces, distributions, ODE) which is very important in analysis and Fourier analysis. $\endgroup$ – reuns Aug 22 '17 at 20:33
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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.

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You can take a look at Linear Algebra of Kenneth Hoffman -Ray Kunze

Here is the link to download it.

http://plouffe.fr/simon/math/HuffmannKunze.pdf

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