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Linear Algebra Seeking Advice on Linear Algebra Textbooks Selection

  1. Jan 6, 2015 #1
    Dear Physics Forum personnel,

    I am a college sophomore with double majors in mathematics and microbiology. I apologize for this interruption but I wrote this email to seek your advice and recommendation on linear algebra textbook. I will be taking the "theoretical, proof-based" introductory linear algebra on upcoming Spring Semester that will not only teach the theoretical linear algebra but also the introduction to proof methodology (this course is required before advancing to analysis and advanced mathematics in my university). The professor will use the online textbook (free) known as "Linear Algebra Done Wrong!" by Sergei Treil (which is free on online). However, I want to purchase the two hardcopy textbooks on theoretical, introductory linear algebra because I always enjoy studying from different textbooks. I am constantly hearing about the books written by Hoffman & Kunze, Friedberg, Axler, Lang, etc. Among those books or others you recommend, which two linear algebra textbooks should I purchase? I usually pick one for the great depth and explanation and another one for more challenging introduction & advanced knowledge. I never took the computational linear algebra before...is it difficult to learn the computational aspect if I only study the theoretical aspect of the linear algebra? Also which book is better to learn the proof methodology, one by Velleman's How to Prove It or Polya's How to Solve It?

    I apologize for asking questions in row and I look forward to your great advice!


  2. jcsd
  3. Jan 6, 2015 #2
    Theoretical and computational aspects of linear algebra are rather different things... Essentially, linear algebra is a set of pretty specific areas, each of them having a set of well-established methods and algorithms. If you envisage that your future occupation/job is likely to involve solving systems of linear equations and/or solving eigenvalue problems, the best way is to take different methods and associated software and begin running test examples, at the same time studying theoretical background of the corresponding methods. Here are a few examples to start from:
    (a) Solving systems of linear equations with dense matrices, using LU-factorization (at the same time you will learn about such fundamental things as the matrix condition ratio, numerical errors evaluation, etc.) Look for "Computer methods for mathematical computations" by Forsythe, Malcolm and Moler (available from amazon.com); it contains a couple of pretty good subroutines: DECOMP and SOLVE.
    (b) Next step: so-called "sparse matrices". They are a common occurrence in numerical methods for solving PDE. The book to look for is " Computer Solutions of Large Sparse Positive Definite Systems" by Alan George and Joseph Lui. Lots of procedures, with solid theoretical background.
    (c) Now to eigenvalues. A good choice would be "The symmetric eigenvalue problem" by Beresford Parlett.
    IMHO, no need to look for specifically "good" theoretical books. Just begin reading the basics and running software at the same time, on a variety of examples.
  4. Jan 6, 2015 #3
    Thank you for the recommendation but those books are quite advance for me and not what I am looking for...
  5. Jan 6, 2015 #4


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    I highly recommend these two. They're both excellent and they offer different viewpoints/emphasis so they complement each other nicely.

    Lang's Linear Algebra (not Introduction to Linear Algebra, which is much more basic)
    Axler's Linear Algebra Done Right

    I just noticed that there's a brand-new third edition of Axler, with substantial improvements and a lot more exercises: http://linear.axler.net/ Or save a few bucks and get the second edition, which is still a very nice book.
  6. Jan 6, 2015 #5
    Thank you very much for the advice! I heard that Axler's linear algebra book is not well suited for the first exposure and I am planning to read that during the summer. I am currently planning to use one from Friedberg or Hoffman&Kunze and Lang's Linear Algebra (not introductory). Between Friedberg and Hoffman, which one complement well with Lang's LA? Can Axler complement Friedberg or Hoffman? I also attached the site that contains the LA etextbook my theoretical la course will use.
  7. Jan 6, 2015 #6


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    sergei treil's book is enough on its own. if you want another book, you may try hoffman and kunze for a good theoretical book, but i do not recommend axler's book, as it is only for a second course. good books for both theory and computations include shilov, or friedberg, insel, and spence. my class notes are free at:


    see notes there for various courses, such as 3000, 4050, 843-4-5, primer of linear algebra,....
  8. Jan 9, 2015 #7
    I would not recommend Lang's linear algebra. His treatment of inner product spaces is atrocious: he does introduce the inner product for arbitrary fields (pointless, because there is no positivity for general fields, and so there is no non-trivial theory), but completely ignores the case of complex spaces. Axler's book can be a good complementary book, after you have learn the basics, but it completely ignores determinate, which, in my opinion, are important. But I probably recommend Hoffman and Kunze over Axler: It is deeper and covers more material, but it is also probably a book for the second course. Friedberg, Insel, and Spence is easier than the Axler or Hoffman--Kunze, but I feel it is a bit dry.

    I sugest to look at "Finite-Dimensional Linear Analysis" by Glazman and Lyubuch. It is not a traditional textbook, but more like a problem book. But the problems (and their order) are carefully selected, so solving several consecutive problems you get a proof of a theorem, often of a highly non-trivial one. So you will be proving all the theorems, not just reading the proofs: the best way to learn any subject, in my opinion!

    It is available as a Dover book, so it is not expensive. The spelling of the authors' names is Glazman and Ljubic in the Dover's edition.
  9. Jan 18, 2015 #8
    I recommend almost any Dover book. Most are older books with new covers but they're great.
  10. Jan 18, 2015 #9
    I like Axler a lot for a first course. I would disagree with those who think it's more advanced. I took that course from Dr. Axler and it was a junior level course intended for those who had finished their calculus sequence. If this is your very first exposure to more abstract, proof based mathematics I would suggest that you think about a "first course" type of book. I know there's one by Smith, Eggen, and St. Andre and there are others. "Reading, Writing, and Proving" by Daepp and Gorkin is also a helpful book for the beginner. I assume you are already familiar with vectors and matrices in R3.
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