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Assuming one wishes to study numerical linear algebra in depth (and follow journals on numerical linear algebra and matrix theory), which would be a better course/better book to take up at first:

With Hoffman and Kunze with proofs and rigor (I don't have problems with rigorous math).

OR

With Prof. Strang's book with un-rigorous proofs or "stated without proof" approach but heavy on applications and "real world" problems.

OR

Any other you would recommend? (How about Gene Golub's book?)

I know some bits and parts of Strang's book (supplemented by his online lectures) and some portions of numerical linear algebra from Trefethen and Bau. But, I wish to have a more thorough understanding of the subject. I will mostly self-study the books.

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I'd probably start with Gil Strang's Introduction to Linear Algebra. It's best to get a solid foundation of the subject without proofs before moving on to a rigorous introduction, like learning calculus before studying real analysis.

After you study Strang's book, if you're still interested in learning more about the rigor behind linear algebra, you could try Sheldon Axler's Linear Algebra Done Right, Halmos' Finite Dimensional Vector Spaces (sort of reads like Rudin), or Mike Artin's Algebra (for more of an abstract algebra take on things; I took his first semester abstract algebra class and loved it). Meyer's book on Matrix Analysis is also supposed to be good.

If you're more interested in numerical linear algebra after that, you could take a look at Trefethen and Bau, Demmel's Applied Numerical Linear Algebra, and Stewart's books on Matrix Algorithms.

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    $\begingroup$ I don't do a lot of research in numerical linear algebra; I know enough of it not to do anything ridiculously inefficiently. My general opinion is that a proof-based course is better if you believe you'll be developing new numerical methods, since you will be expected to prove that your methods work if you submit to a math journal, and if you don't submit to a math journal, you should still prove that your methods work. If you're not developing new numerical methods, then you probably don't need that level of rigor, even though it "builds character." $\endgroup$
    – Geoff Oxberry
    Jan 28 '12 at 18:26
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    $\begingroup$ Excellent list, Geoff. Another bump for Trefethen & Bau, and if you happen to be working in sparse matrices/partial differential equations, Iterative Methods for Sparse Linear Systems is a gem. $\endgroup$
    – Aron Ahmadia
    Jan 28 '12 at 20:03
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    $\begingroup$ True. Hard to ignore Saad when it comes to Iterative Solvers or NLA in general. $\endgroup$
    – Inquest
    Jan 28 '12 at 20:09
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    $\begingroup$ In response to "Is a proof based course necessary?" -- You don't need to be able to prove things but I think it's crucial to get a more-than-numerical understanding of LA. An abstract coordinate-free view of vector spaces and linear transformations can be extremely helpful in understanding problems. $\endgroup$
    – MRocklin
    Jan 29 '12 at 3:33
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    $\begingroup$ @MRocklin Agreed. Strang's book is probably the closest one can get to that without having to prove something. $\endgroup$
    – Geoff Oxberry
    Jan 29 '12 at 3:44
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I "grew up" with Golub & Van Loan. In my opinion, best book for both theory and implementation.

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  • $\begingroup$ Would you recommend Golub as the first LA textbook a student ever touches? $\endgroup$
    – Inquest
    Jan 30 '12 at 12:27
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    $\begingroup$ In principle, it could be, but in practice, G&VL doesn't go into enough details on the basics of linear algebra. There's too much left unsaid to make it the only LA text a person sees. $\endgroup$
    – aeismail
    Jan 30 '12 at 14:13
  • $\begingroup$ @Nunoxic: it was my first and I survived :-) But we had a great teacher who maybe filled the gaps unnoticeably... $\endgroup$
    – GertVdE
    Jan 31 '12 at 19:36
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G. H. Golub and C. F. Van Loan, Matrix Computations, third edition, The Johns Hopkins University Press,Baltimore, 1996.

N.J.Higham, Accuracy and Stability of Numerical Algorithms, SIAM, 1996.

Y.Saad, Iterative Methods for Sparse Linear Systems, SIAM, 2000.

L.N.Trefethen and D.Bau,III, Numerical Linear Algebra, SIAM, 1997.

H. A. Van der Vorst, Iterative Krylov Methods for Large Linear Systems, Cambridge University Press, 2003.

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