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Advice on Books?

  1. Apr 23, 2010 #1
    I'm thinking of doing the following to actually learn linear algebra thoroughly as opposed to the 50 page treatments a vector calculus book will offer. Could you offer any advice on whether or not the following plan is reasonable to do on my own?

    1: Do all of the 34 video lectures on the mit page with "Linear Algebra & it's Applications" 4ed.

    The book I can get cheaply (in the link below) is not exactly the same one that the website advises but the contents look extremely similar and it's by the same author, the chapter contents mirror the lectures pretty closely.

    http://ocw.mit.edu/OcwWeb/Mathematics/18-06Spring-2005/VideoLectures/index.htm
    https://www.amazon.com/Linear-Algebra-Applications-Gilbert-Strang/dp/0030105676/ref=ntt_at_ep_dpt_6


    2: Introduction to Linear Algebra by Lang

    to get into the proofs.

    https://www.amazon.com/Introduction...=sr_1_2?ie=UTF8&s=books&qid=1272013454&sr=1-2

    3: Linear Algebra Done Right by Axler

    To master the proofs!

    https://www.amazon.com/Linear-Algebra-Right-Sheldon-Axler/dp/0387982582/ref=pd_cp_b_2

    4: Linear Algebra by Lang

    The graduate level book (I think).

    https://www.amazon.com/Linear-Algeb...=sr_1_1?ie=UTF8&s=books&qid=1272013454&sr=1-1

    Thanks I'd appreciate a comment or two if you've used any of these books especially.
     
  2. jcsd
  3. Apr 23, 2010 #2
    I haven't used any of those books. I would suggest however, that "Abstract Algebra" by Dummit and Foote is in my opinion the best text money can buy for learning any kind of basic algebra. I just wish I knew that when I was learning it!
     
  4. Apr 23, 2010 #3

    Landau

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    Well, if you do all this you'll certainly know linear algbra well. But it may be a bit too much; there's quite some overlap between the books. I think if you do 1 (with Strang's book), you won't need 2 (Lang's intro). And if you do 3 (Axler), you won't really need 4 (Lang's other book).
    If you do Strang, and then either 3 or 4, you'll be great. Also, Lang's second book isn't graduate level. Both Axler and Lang's 2nd are more like "second linear algebra books".

    If you want to continue, there's the great Advanced Linear Algebra by Steven Roman, which is a graduate level text (at least it's a GTM). But you'll have to know some algebra (he works with modules).
     
  5. Apr 25, 2010 #4

    jasonRF

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    "Introduction to linear algebra" by Strang is, in my opinion, easier to learn th subject from than his "linear algebra and it's applications". That is why it is the book actually used in the mit course these days, and if you are going to spend all those hours watching hte lectures and working homework problems, you might be better off with "introduction to ...". Both books do a great job of presenting a matrix-centric approach to the subject that includes important applications. I own "linear algebra and its applications" and refer to it often.

    Depending upon what you hope to get out of your study of linear algebra, that might be all you need for a first go-around! I must say that when I self-study a topic, I often have plans of working multiple books at a couple levels, but my interest usually dies by the time I finish the first book.

    If you decide you want a more theoretical approach after you finish Strang, the book by Axler is a nice choice. I worked through most of it on my own and find that it complements Strang very well. Almost all of hte end of chapter problems are proofs, and I thought it was a fun challenge to work a bunch of them.

    Of course, whatever book(s) you work through, seriously attempting a lot of problems is essential.

    good luck,

    jason
     
  6. Apr 25, 2010 #5
    Thanks for the answers guys, in my head I would see Strang + the lectures as being more computational. I know a lot of it but forget it so I'm just going to do all the lectures & master this book to 100% get beyond forgetting these stupid computational methods.

    I would see the first Lang book as being a good intro to the theory & proofs in linear algebra, but if you think it's low on the proofing maybe I should just wait for Axler.
     
  7. Apr 25, 2010 #6

    HallsofIvy

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    But Dummit and Foote's "Abstract Algebra" does NOT cover Linear Algebra which is what is being asked about.
     
  8. Apr 25, 2010 #7
    Last edited by a moderator: Apr 25, 2017
  9. Apr 26, 2010 #8

    Landau

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    Altough D&F is not really good to learn LA (at least for the first time), it does cover Linear Algebra: Chapter 11 is called "Vector Spaces", after Chapter 10 about Module Theory.
     
  10. Apr 26, 2010 #9
    Landau, the way you are speaking about Dummit & Foote makes abstract algebra sound easy but it's not such an easy subject. I was under the impression that you needed to be extremely capable with proofs and at least be studying analysis concurrently. Linear Algebra seems to be the far easier subject compared with abstract algebra, unless I'm mistaken.
     
  11. Apr 26, 2010 #10

    Landau

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    How do I make it sound easy? No, you certainly need to be comfortable with proofs, and some basic number theory (analysis won't really help, except for the 'mathematical maturity', familiarity with proofs, you'll gain from it). Linear algebra as it is usually taught in a first course is indeed 'easier' than abstract algebra, focussing on matrix computations and some theorems. But formally, linear algebra is just a subbranch of abstract algebra, and from a higher perspective linear algebra is just a special case of module theory (a vector space is just a module over a field).
     
  12. Apr 26, 2010 #11
    This is the book I used when I took Linear Algebra,
    http://www.math.brown.edu/~treil/papers/LADW/LADW.html
    "Linear Algebra Done Wrong" by Serge Treil.

    I personally feel it is a well written introductory level text on Linear Algebra. It is also less computational than similar books at an intro level of Linear Algebra. So you will be doing proofs throughout the book, as well as reading them. If you are interested in going through a semester worth of L.A., then cover the first seven chapters in the book mentioned above. It is clear, and at the beginning it gives a more concrete approach with more examples which become sparse as you dwell deeper into the theory.

    As for the suggestion of Dummit and Foote, while a great book, I do not recommend it as a means of learning L.A. for the first time. Although once you are done teaching yourself L.A. I would suggest using Herstein's Topics in Algebra to teach yourself Abstract Algebra (and Herstein does a wonderful coverage of L.A. in Ch 4).

    Good luck in your studies.
     
  13. Apr 26, 2010 #12
    Thanks for the suggestion, I actually found that pdf online the day I was planning out my LA future but while it looked pretty good (and had a snazzy title :wink:) I think I'm just better off sticking to a book with exercises & answers until I've come fully to terms with LA proofs beyond just finding an inverse :rofl:

    Unfortunately, Unlike Jason I have a compulsion to finish the books I start, It's a bit ridiculous actually but I'm still trying to include finishing off all of the calculus videos http://www.math.armstrong.edu/faculty/hollis/calcvideos/ for closure, (the same as the link above for linear algebra and differential equations!).

    LA Done Wrong has stuck in my mind though and if you'd just randomly suggest it out of all the possibilities I might give it a proper looksee once I've at least finished the Strang videos.

    Also, I've heard a lot of good things for Fraleigh's Abstract Algebra as a first course, but I'll check the reviews out for your suggestion too, merci :biggrin:
     
  14. Apr 26, 2010 #13
  15. Apr 27, 2010 #14
    So here is my suggestion, if you are planning to pursue Mathematics as your degree in college or beyond, challenge yourself now and struggle on the proof based books. It is better to get used to it now, for almost every book beyond LA does not have solutions at the book of the book nor a solutions manual. Another course of action would be to go through a book specifically meant to learn how to write proofs, as in that manner you will become more familiar with how to prove things in math and it will make your future studies easier to some extent. As for Fraleigh's Abstract Algebra, I feel that it simplifies topics too much. For instance, Sylow's Theorems are considered in his book "advance group theory" and as a result he does not cover it until the end of the book. Yet in an undergraduate class you will cover them when you go over group theory. But your call, I would say if after LA you feel confident with proofs to go with either Herstein or Dummit and Foote. Herstein writes the book in a friendly manner as if he were right there guiding you, however his notations are annoying at first. Dummit and Foote is verbose and filled with many examples. So enough of my ranting, good luck! and remember not to be afraid of challenging yourself.
     
  16. Apr 28, 2010 #15
    I would say start with Axler's book. Axler gives a clean presentation of linear algebra, there's no un-necessary fiddling with matrices or any other numeical and number-crunching nonsense. You will see the heart of linear algebra, the essential structure of the subject without being drowned in a see of indixes or matrices. The book is exceptionally well written.

    However the book is not comprehensive and doesn't deal as much with matrix-crunching as other books do (most linear algebra courses go the matrix-crunching route). The latter point is not a drawback-- it is a plus actually-- it's just that you're expected to know all the matrix-crunching stuff. I don't know what would be a good book to follow Axler. Maybe Shilov's book.

    In short:
    Start with Axler. You will see clearly what linear algebra is about, with the structure and beauty of the subject not being lost in icky matrix calculations.
    Follow it up with Shilov, who presents linear algebra from a very different perspective, and covers topics Axler misses.
    Those two books should be enough for an undergrad.
     
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