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I'm currently taking a numerical analysis course. We are covering linear algebra topics, the gist of the first chapter of the course being solving systems of linear equations. The lecturer has introduced SVD decomposition, condition number of a matrix, LU decomposition and QR decomposition (using Householder decomposition).

In the past I've been used to very rigorous, thorough and well-organized math courses. But I feel like what I'm learning here is shallow since there are very few proofs.

So my question is

Can you recommend rigorous textbooks or detailed lecture notes that cover the topics I mentioned above ?

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    $\begingroup$ Though I'm sure others can recommend such textbooks, I'd like to caution against the attitude that proofs are the only approach to understanding mathematics deeply. Though this can even be true in certain parts of pure mathematics, it is especially true in applied mathematics such as numerical analysis. In this subject, having a thorough understanding of the behavior, scope, and drawbacks of the main computational algorithms is much more important than being able to place these algorithms into a rigorous theoretical framework. $\endgroup$ – Jim Belk Jan 21 '16 at 15:49
  • $\begingroup$ You can check out the textbook recommendations in the following MSE post. $\endgroup$ – Jose Arnaldo Bebita-Dris Jan 23 '16 at 11:03
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The following books cover all the topics that you mention rigorously. They are required reading in the field of numerical analysis and require a solid background in pure mathematics.

Higham, N. J: "Accuracy and stability of numerical algorithms", SIAM, (2002)

Golub, G. H and van Loan, C. : "Matrix computations", 4th edition. John Hopkins University Press, (2013)

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At the level you're talking about, I think a good choice is Trefethen and Bau, Numerical Linear Algebra. It covers the topics you mention in reasonable depth while still being very approachable.

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