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I am almost finishing Gilbert Strang's book "An introduction to linear algebra" (plus video lectures at MIT OCW). First and foremost, I would like to suggest this course for everyone. It has been incredibly illuminating.

I would like to continue studying linear algebra, with particular focus on different properties of matrices and the transition to more general linear spaces (I am a physicist so Hilbert spaces and etc. are of particular interest).

Does anyone have a a good recommendation of books/resources/etc.?

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Hoffman and Kunze? – Inquest Jul 21 '12 at 0:10
@Inquest, H&K is probably too abstract. It's a nice book, though. – lhf Jul 21 '12 at 0:20
As a warning, I have seen a textbook on infinite-dimensional linear algebra advise that one forget everything one has learned about finite-dimensional linear algebra, as any intuition you have for the subject it is just as likely to be misleading as it is to be helpful. – Hurkyl Jul 21 '12 at 0:27
wikipedia is actually pretty good, and you can further reading references. – chaohuang Jul 21 '12 at 3:55
How about reading the chapters on vector spaces and linear transformations from I. N. Herstein's Topics in Algebra. Don't forget to do the exercises. – Shahab Jul 21 '12 at 10:08
up vote 7 down vote accepted

I have to suggest the somewhat underrated Matrix Analysis by Horn and Johnson (the first edition was used for my ALA class at NCF.) They take a wonderfully concrete approach to most topics encountered in a second linear algebra course (Schur Decomposition, Spectral Theorem for Normal Operators, Jordan Canonical Form, Singular Value Decomposition) while adding a lot of other nice things into the mix. The fourth chapter on Hermitian Matrices talks about the Rayleigh Ritz Theorem and variational characterization of eigenvalues, which I imagine come up a lot in serious study of classical mechanics. Chapter five discusses finite dimensional inner product / normed / pre-normed spaces in terms of algebraic, analytic, and geometric properties. They include a discussion of completeness and the $l^p$ norms, which I guess could be seen as a preview of Hilbert Space Theory. There are also nice sections on the Gersgorin circle theorem and numerically solving linear systems.

I think it's a wonderful choice for any student, but especially a non-mathematician. The proofs are rigorous and sometimes tedious but always understandable. Typically, things are proved in an algorithmic fashion rather than through diagram chasing or algebraic artifice (nary a mention of finitely generated modules over a principal ideal domain.) My only complaint is that there are a fair number of results assumed regarding matrix algebra and determinants which wouldn't typically appear in a linear algebra course - references for these are typically not too hard to find though.

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You could try Meyer's Matrix Analysis and Applied Linear Algebra or Lax's Linear Algebra and its Applications.

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I took a "second semester" course in linear algebra out of Friedberg, Isnel and Spence's book Linear Algebra and found it to be a good read and a really useful reference.

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An old edition of Friedberg, Insel & Spence along with this video course which follows the book.

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It may also be that, for applications (whether within or without mathematics proper) the label "linear algebra" is misleading. I'd think that future physicists might care more about (positive-attitude) "functional analysis" than what the traditional mathematics enterprise calls "linear algebra". Depends.

There is also the conflict, alluded-to in comments and answers, that the orthodox/traditional math texts "advancing" linear algebra are a bit aggressive in demanding compliance-in-orthodoxy from the reader. Contemporary math-phys or phys students may bridle at this, and reasonably so, considering that it is mostly due to the typical curricular/textbook inertia.

E.g., the earlier remark that people are abjured by (presumably orthodox) authors/authorities to "forget about finite-dimensional linear algebra to address infinite-dimensional" disserves everyone (even if it can be interpreted as being "true" in some way). That is, while the naive-est extrapolation of finite-dimensional linear algebra proves inadequate, it is nevertheless the model for what everyone (if they were candid) would like to be true. Thus, the task partly becomes appraisal of how far from this ideal we find ourselves in any particular real-life situation.

The main "life-lesson" is about "continuous spectrum", which is exemplified by Fourier transform and Fourier inversion on the real line. On one hand, "everything is fine", but, when juxtaposing to (perhaps naive) linear algebra, there are problems. But, in fact, it truly is fine, when one adjusts one's notion of the range of happy, useful, extensible answers.

Peter Lax's functional analysis book is certainly vastly more sophisticated than upper-division linear algebra books, but th'guy is very practical-minded, and is not adversarial as a writer. His interests in PDE are physical.

My own notes about functional analysis and linear algebra, at my web page, while not pretentious or anti-physical, are written from a viewpoint which may not be the most useful to the questioner, though, who knows? At least I tried to be honest. :)

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I recommend Halmos' Finite Dimensional Vector Spaces. Halmos was a functional analyst and an expert on Hilbert spaces. I read that the goal of the text is to present finite dimensional vector space theory as the "easy case" of Hilbert space theory.

Added later: You might also check out Janich's book. I noticed it had problem sections titled, "For physicists."

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I should note that the book is fairly abstract, though. – user59083 Jul 27 '13 at 16:36

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