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I plan to self-study linear algebra this summer. I am sorta already familiar with vectors, vector spaces and subspaces and I am really interested in everything about matrices (diagonalization, ...), linear maps and their matrix representation and eigenvectors and eigenvalues. I am looking for a book that handles every of the aforementioned topics in details. I also want to build a solid basis of the mathematical way of thinking to get ready to an exciting abstract algebra next semester, so my main aim is to work on proofs for somehow hard problems. I got Lang's "Intro. to Linear Algebra" and it is too easy, superficial.

Can you advise me a good book for all of the above? Please take into consideration that it is for self-study, so that it' gotta work on its own. Thanks.

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    $\begingroup$ get Lang's Linear algebra $\endgroup$
    – Artem
    Commented May 22, 2014 at 2:39
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    $\begingroup$ @user1551 Lang's Intro. to Linear Algebra and Linear Algebra are different. $\endgroup$
    – fkraiem
    Commented May 22, 2014 at 7:02
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    $\begingroup$ Perhaps old fashioned but I learned from "Finite dimensional vector spaces" by Paul Halmos. $\endgroup$
    – Ragib Zaman
    Commented May 22, 2014 at 8:38
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    $\begingroup$ I strongly recommend you get one of the following matrix-based programming languages: Matlab or Octave or Julia. Then play around/experiment with matrices as you go. Actually compute and manipulate the matrix factorizations that you read about in a theorem. Make and test your own hypotheses, then try to prove them. I learned more about linear algebra this way than I did in any of the classes I took or books I read. $\endgroup$
    – Nick Alger
    Commented Jul 4, 2016 at 4:15
  • $\begingroup$ Does this answer your question? What is a good book to study linear algebra? $\endgroup$
    – user1147844
    Commented Apr 12, 2023 at 6:01

12 Answers 12

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When I learned linear algebra for the first time, I read through Friedberg, Insel, and Spence. It is slightly more modern than Hoffman/Kunze, is fully rigorous, and has a bunch of useful exercises to work through.

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    $\begingroup$ Thanks, I conducted some research and found it could be what I am exactly looking for. What about the difficulty of exercises? $\endgroup$
    – Mike
    Commented May 22, 2014 at 11:22
  • $\begingroup$ I do not think I found them very difficult, but it was a long time ago. Besides, you may find it more or less difficult depending on your background. $\endgroup$
    – Christopher A. Wong
    Commented May 22, 2014 at 18:25
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A great book freely available online is Linear Algebra Done Wrong by Sergei Treil. It covers all the topics you listed and culminates in a discussion of spectral theory, which can be considered a generalized treatment of diagonalization.

Don't be put off by the book's title. It's a play on the popular Linear Algebra Done Right, by Sheldon Axler. Axler's book is also very good, and you might want to check it out.

The classic proof-based linear algebra text is the one by Hoffman and Kunze. I find the two books I listed above easier to read, but you might also consider it. In any case, it is a good reference.

I hope this helps. Please comment if you have any questions.

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  • $\begingroup$ Thanks for your consideration. I didn't know about Treil's but I did for Axler and it seems like it overemphasizes the computation part of the topics. Anyway, I will look at it thoroughly. $\endgroup$
    – Mike
    Commented May 21, 2014 at 23:30
  • $\begingroup$ @Mike Well, computational facility is very important to develop! But you can always just skip the overly parts if you find them tedious and feel you don't need any more practice. It's self-study, after all. $\endgroup$
    – Potato
    Commented May 22, 2014 at 0:30
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Linear Algebra by Hoffman-Kunze. Might be a little too deep, but I believe you'll do fine with it.

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  • $\begingroup$ It already deals with abstract algebra, which I'm not yet familiar with. I went trough some books and I found Axler's too computational and a bit superficial. Hefferon's looks somehow good. $\endgroup$
    – Mike
    Commented May 21, 2014 at 23:23
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    $\begingroup$ Axler is too computational and superficial? Are you actually doing problems and re-working proofs, or are you just skimming the text? $\endgroup$
    – Batman
    Commented May 22, 2014 at 1:08
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    $\begingroup$ Axler is a good read, but it's more suitable for reflections on than for introduction to the subject. The main theme of the book is to criticise the (ab)use of determinant. While the author does have a point (that's why I think the book is good for reflections), his "det-phobia" has some undesirable consequences. E.g. (IIRC) in order to avoid using determinants, he has to define characteristic polynomials for real and complex matrices differently. I don't see how abandoning a unified approach is "linear algebra done right". $\endgroup$
    – user1551
    Commented May 22, 2014 at 4:23
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Jim Hefferon at Saint Michael's College, has a pretty well known linear algebra textbook that he provides for free: Linear Algebra.

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old, but good: Linear Algebra and its Applications by Gilbert Strang, see http://www.amazon.com/Linear-Algebra-Its-Applications-Edition/dp/0155510053/ref=sr_1_4?ie=UTF8&qid=1400721854&sr=8-4&keywords=strang+algebra

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Linear Algebra and Its Applications, 4e by David C. Lay

This is the #1 rated Linear Algebra book on Amazon. It should be good! I'm using it for a class next semester here at UW, which is ranked #9 in the country for Mathematics.

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    $\begingroup$ I do not think that Lay's book covers the subject in the level of detail that the OP is looking for. $\endgroup$
    – Christopher A. Wong
    Commented May 22, 2014 at 6:28
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I can't believe nobody has mentioned Peter Lax's Linear Algebra and Its Applications.

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I can recommend you two books. One already mentioned by @JPi and it is Linear Algebra and Its Applications by G. Strang: http://www.amazon.com/dp/0155510053/?tag=stackoverfl08-20

Another book that I like very much is Fundamentals of Matrix Computations by D. S. Watkins: http://www.amazon.com/Fundamentals-Matrix-Computations-David-Watkins/dp/0470528338 You can find there fundamental algorithms that are used in the field of matrix computations.

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Carl Meyer's Matrix Analysis and Applied Linear Algebra is my favourite. While many of the above books are good, Meyer's book has a great focus on how you actually find the various objects - this is in stark contrast to Hoffman-Kunze who are content with existence proofs.

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If you are new to linear algebra, then you should use "Introduction to Linear Algebra" by Gilbert Strang. In case you possess some knowledge of LA, then you can use "Matrix Theory and Linear Algebra" by I.N. Herstein. There are many books on pure linear algebra and computational linear algebra; you can choose as per your requirement and interest.

The two books I have recommended can serve as foundation for both.

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Linear Algebra by Fraleigh is a good book.

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S. Winitzki, Linear Algebra via Exterior Products (free book, coordinate-free approach)

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