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First exposure to Linear Algebra

  1. Sep 25, 2014 #1
    After having covered single variable calculus to a rather thorough degree, I would now like to move forward to linear algebra. As such I would like to enquire as to any recommendations for a text appropriate for what is essentially a beginner to the subject (I have received very basic computational courses involving matrices). The long term aim is to cover an introduction and then further cement the subject by working through Axler's Linear Algebra Done Right, so any text which fills in the gaps of Axler's work whilst also setting up well for it would be preferable. My further long term aim is to then work through Spivak's Calculus on Manifolds (although I am open to other multi-variable calculus text suggestions, especially to supplement or serve as preparation for said text). After this I do intend to cover some quantum mechanics, so texts which treat topics important to this area well are desirable, however I definitely would prefer to cover any maths required for this in the fashion of a mathematician as opposed to working through a "maths for physicists" style text. For full disclosure I have covered single variable calculus by first working through Lang's "A First Course in Calculus" and then following through with the text by Spivak. Although I purposefully excluded the section on multi-variable calculus in Lang's book, deferring it for later study. I am currently considering between about 3 texts for my intro to Linear Algebra, those being Lang's "Introduction to Linear Algebra"; Anton's "Elementary Linear Algebra (7th edition I do believe); and Sergei Treil's "Linear Algebra done Wrong". Of course I am still open to other options on this front (both Shilov and Hoffman & Kunze are attractive prospects but appear to be written at a higher level than would currently be apt). Finally I must confess that, despite enjoying his Calculus text, on my first skim through chapter 1 of Lang's "Intro to Linear Algebra" I found the writing style employed to be rather dry.

    Thank you for any help you may offer and apologies for the wall of text. :D
  2. jcsd
  3. Sep 25, 2014 #2


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    mit open courseware has a linear algebra series. Edx also had a linear algebra course, through ut austin. I tried to do it, but it was difficult for me. I was trying to learn linear algebra and learn to program with python at the same time. <-- used python for linear algebra.
  4. Sep 25, 2014 #3
    I actually have copies of all three of Lang, Anton and Treil, but I have only looked at Lang in detail. Anton is not so good, I don't recommend it. I would say either Lang or Treil, but I really like Lang as an introduction, as it's quite visual, an it has answers/solutions/hints at the back. Anyway, Treil's book is available for free from his website:
    If you think Lang's Intro to Linear Algebra is dry, just wait til you see some of his other books. But seriously, I think it's a very good introduction. Maybe you think it's dry because it uses formal proofs? If so, that's just something you have to get used to. Anyway, use both.
    After that go on to Abstract Algebra, for which Artin is highly recommended, and use Axler as a supplement to that. If you still feel like more Linear Algebra, then go for Shilov, or Hoffman and Kunze if you have the money.
  5. Sep 25, 2014 #4
    Lang's good but dry, quite boring and has some mistakes (a few too many, but maybe my edition is just especially old. Treil is lively and fun; great book, but more advanced (maybe read them side by side?). Maybe you're ready for Axler right away, it's a pretty easy book... It's not very computational, but it's real linear algebra (Lang is just the opposite, and Treil is somewhere in between). It's great for becoming acquainted with proofs as well (although you're probably great at proofs already since you did Spivak). I'd say Axler is *much* easier than Spivak, so if you were comfortable with that, go for it. You'll have to supplment Axler with another text (Treil is good for that) to learn matrices more thoroughly (Axler's book is more abstract, it skips the matrix treatment until the very end; like I said, real linear algebra. That's why Treil 'does it wrong').
  6. Sep 26, 2014 #5
    If you're learning linear algebra before you study Calculus on Manifolds then start with Lang. Axler won't cover the matrix and determinant stuff you need at the beginning. I can't agree with the 'dry', 'boring', 'full of mistakes' comments from guitarphysics about Lang. Lang is also 'real' linear algebra, whatever that means, it's just a more traditional approach, which doesn't mean it's 'fake'. I high recommend you start with Lang, then move on to Treil and Axler. In fact, I don't think you'll really appreciate the beauty of Axler unless you study Lang or Treil first.
  7. Oct 6, 2014 #6


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