I'm a physics major and I'd probably go with mathematical/theoretical physics path.

Where do I start with self learning linear algebra? I'm good with proofs but I'm not comfortable with learning math without intuition or motivation behind the axioms. Still, I hate math without rigor (cookbook engineer math).

I'm looking for an intro book for linear algebra. Thanks. May be my first exposure to pure math excluding intro to proofs.

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There are a lot of good online resources. Here's one: Hefferon's book: joshua.smcvt.edu/linearalgebra , which is linked to Sharipov's Quick Introduction to Tensor Analysis: math.stackexchange.com/a/67392/8157 . – Giuseppe Negro May 16 '12 at 17:41

There are many, many ways that this question could possibly be answered. If, as you say, you are really comfortable with proofs, and you want a very rigorous approach, you might consider learning linear algebra from an actual "algebra" book instead of a "linear algebra" book. I think my mistake when (re)-learning linear algebra is not taking this route. Not all "algebra" books have good coverage of linear algebra though some that do include:

All of these books are readable and have very good coverage of linear algebra. Of course, if you go this route, you will end up learning more standard algebra topics than just "linear algebra". One could argue though that such standard topics are needed anyway. Consider, for example, the standard definition of the determinant via the classical combinatorial formula. To really make sense of this and to work with it effectively you need to know about permutation groups and various facts about permutations such as every permutation can be expressed as a product of transpositions. These facts really are needed in linear algebra but most linear algebra books don't explain them thoroughly. On the other hand, every good algebra book will.

On the other hand, if you want something that is theoretically sound but more elementary and/or computationally oriented, you might consider one of these:

Both of these texts are very readable and have a good mixture of theory/application. One thing nice about Meyer's book for self-study is that complete solutions are provided for the exercises. Also, all of the above books can be used profitably together though don't expect the same topics to be treated the same way.

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@ItsNotObviousThanks for the Module Theory suggestion. I was looking for an good source. – Andrew May 16 '12 at 20:41
@Andrew If you're interested in the module theory angle, there's another book that I've mentioned here before that has a similar flavor; it's by T. S. Blythe and its called "Module Theory: An Approach to Linear Algebra" – ItsNotObvious May 16 '12 at 20:43
@ItsNotObviousThanks – Andrew May 16 '12 at 20:47
@ItsNotObvious: Aren't these grad level books? – Ron Jan 16 '13 at 14:36
I second the recommendation of Artin, Algebra. I learned linear algebra (very well) from self-studying this book. (I had some previous knowledge of matrix mechanics but no rigorous or theoretical linear algebra knowledge before I picked it up.) – Ben Blum-Smith Apr 25 '13 at 18:58
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Strang is an endearing lecturer, but this course is more aligned toward matrix mechanics. In my personal opinion, if you are inclined toward math and theoretical physics, you would benefit from Axler (see below), – Andrew May 16 '12 at 19:54
@Andrew : Thank you for the info, I wish there was a list of lecturers and a brief review and recomendation like your comment, that would help everyone immensly. – Arjang May 16 '12 at 20:20

I am a self-studier. I have found Axler's "Linear Algebra Done Right" (if you can get past the title) to be very useful. It is thorough (not left to the reader) in its presentation of the material. It's used as a text at Berkeley, MIT, and Harvard. I emphasize its accessibility.

http://math.mit.edu/~trasched/18.700.f11/index.html

And Berkeley:

http://math.berkeley.edu/~scanez/courses/math110/Home/Home.html

http://math.berkeley.edu/~mhaiman/math110-spring12/

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Linear Algebra Done Right by Sheldon Axler. Excellent book for the first two courses in Linear Algebra.

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Oops. Thanks Andrew.. :-) – Amihai Zivan May 16 '12 at 19:47
No problem, Ross has lots of books that are used in similar courses. – Andrew May 16 '12 at 19:50

Mix and match:

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You might try Strang, "Introduction to Linear Algebra". http://math.mit.edu/linearalgebra/ Strang is especially good at explaining ideas and motivation, without sacrificing rigour. You can also see Strang in action at http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/

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Strang is a great teacher but the Videos are somewhat poor in proofs, formalism etc. – Amihai Zivan May 16 '12 at 19:42