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Since this does not appear to have been asked here before, I would like to solicit suggestions and recommendations, ideally with rationales, for a textbook in a first course in linear algebra. In my university, students will not necessarily have taken any proof-based course (we do not have an introduction to proofs course). Our course is expected to cover basic computational issues including Gaussian elimination and eigenvectors/eigenvalues. We also cover theoretical material including an introduction to vector spaces over $\mathbb{R}$ (as well as over $\mathbb{C}$ to deal with diagonalization) culminating, ideally, with a statement and proof of the Spectral Theorem. We expect students to learn how to carry out fairly straightforward proofs that primarily follow from the definitions or directly from one of the theorems covered in class. Because we have a large number of engineers and other science majors in the course, I do plan to spend some time on practical issues as well. I like to discuss rotations and reflections in $\mathbb{R}^3$ using an eigenvector/eigenvalue analysis. I will definitely cover the singular value decomposition and would like to spend some time discussing modified Gaussian elimination to handle numerical problems due to rounding error, though the latter is likely to be omitted because of time constraints.

I've tried several different books before (Strang, Anton) and yet to have found one that had a presentation close to what I have in mind for the course. Any suggestions for books whose aims are aligned with the goals above would be appreciated.

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    $\begingroup$ Do you think it would be worthwhile to change your question to suggestions and rationale? I don't doubt that respondents are earnest in their suggestions (which is precisely what you have asked for) but it would be nice to know about posters' reasons/thoughts... $\endgroup$ – Benjamin Dickman Oct 8 '15 at 8:21
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Hefferon is a wonderful book, and it's also free: http://joshua.smcvt.edu/linearalgebra/ .

In general, I think it's a bad idea to write down a list of topics you want and then look for a book that covers them all. If the topics are standard, this is a waste of time, because they'll be covered in every book. If the topics are not standard, then you're shooting yourself in the foot by ruling out almost all possible books. If you have a few nonstandard topics you want to do, just write up lecture notes on them and put them online.

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    $\begingroup$ Thanks, I appreciate the good suggestions and I agree with your point. I was largely giving a list of topics primarily to try to convey as best as possible the general approach I would like to take. In part, my problem is that I would like to do a little bit of everything, yet still have the students come away with a reasonably strong foundation including an ability to write simple proofs ... and in past experience there has not been enough time to achieve all of this. $\endgroup$ – Michael Joyce Oct 7 '15 at 22:46
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    $\begingroup$ @MichaelJoyce, if there isn't enough time, don't force it. It will just blow up around your ears. "Write simple proofs" is something that takes time (of the calendar variety, not of the grind through exercises variety). Perhaps string it out over a sequence of courses? $\endgroup$ – vonbrand Oct 7 '15 at 22:49
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"Introduction to Linear Algebra" by Professor Gilbert Strang

along with the accompanying video lectures

http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/

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Look at Treil's "Linear Algebra done Wrong", it's free.

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I think Lay's and Poole's books are easily the best of the common ones on the market. However, if you have stronger students than we do, Strang or Bretscher might be more appropriate. (Not having a introduction to proofs class is frequently an indication of stronger students.)

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    $\begingroup$ Could you maybe add links? I don't think I've heard of any of these. $\endgroup$ – Jessica B Oct 8 '15 at 7:20
  • $\begingroup$ I don't really want to endorse Amazon or any other bookseller, and those are the only links I know of. $\endgroup$ – Alexander Woo Oct 12 '15 at 21:58
  • $\begingroup$ How about including titles (and perhaps publishers) then? $\endgroup$ – Jessica B Oct 13 '15 at 6:29
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    $\begingroup$ David Poole's text has ISBN 9780538735452 -- Lay's has ISBN 032198238X Both of these are ridiculously expensive but decent texts. $\endgroup$ – Zach Haney Oct 16 '15 at 16:29
  • $\begingroup$ if you can use an older edition, that brings down the price dramatically. We use Lay where I teach, and I've worked with friends out of that book, using the previous edition. $\endgroup$ – Sue VanHattum Oct 19 '15 at 0:20
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I would back up @Alexander-Woo for Poole's text as well as offer the suggestion for

Shifrin's "Linear Algebra - A Geometric Approach".

None of these books are really affordable. There was a suggestion earlier of a free text, and I say... supplement that, frankly. More work, but it's worth considering your students and their finances at this point.

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