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Advanced Linear Algebra Book Recommendation

  1. Nov 30, 2012 #1
    I am currently a first year graduate student in math, and I am trying to pick a linear algebra book to work through during the winter break. I have already gone through the computational style linear algebra, and I have also gone through Axler's Linear Algebra Done Right. I would like to go through a more advanced LA text, and I have been considering two choices: Halmos' Finite Dimensional Vector Spaces and Steven Roman's Advanced Linear Algebra.

    Roman's looks like it covers more material, and also covers modules and PIDs which would be a welcome review from my Abstract Algebra class this fall. Halmos is, however, a classic, and I'm not sure which would be more appropriate for my eventual goals in differential geometry - both cover the tensor product, although I think perhaps Roman's book does more with it. There's also Hoffman and Kunze, but I looked over the table of contents and didn't see anything I hadn't already covered. Any suggestions?
  2. jcsd
  3. Nov 30, 2012 #2
    I haven't actually read either book, but I checked out their amazon reviews so you can judge my remarks accordingly.

    To me Roman's book looks more interesting and a longer lasting reference, but 544 pages is a lot of pages for winter break. Halmos' is a relatively short classic, so it would probably make good reading over the holiday, but would you keep it on your shelf as a reference later on? I am guessing no.

    That said, for differential geometry, you really only need to know about tensor products (since you already know the content of Hoffman/Kunze), and that isn't too much to learn. If you understand bilinear products and the determinant, then the toughest algebraic aspect of tensors might just be the notation (Einstein notation took awhile to click for me). You might be better served by simply familiarizing yourself with the basics of tensor products and then moving on to Lie Groups / Lie Algebras. Stillwell's Naive Lie Theory looks interesting (you can read the preface on Amazon).
  4. Nov 30, 2012 #3
    If the choice is between Halmos and Roman, then I would certainly go for Roman. It's truly a gem and it covers a lot of cool stuff. You think you know most of Halmos already, so you're probably wasting your time with that book.
  5. Nov 30, 2012 #4
    I'm a big fan of Roman, and seeing as you know almost everything in Halmos from Axler (if not then the first few chapters of Roman will clear that up), there is pretty much no point to reading it, as Micromass said. Greub's 'Multilinear Algebra' is also a good companion for tensors and whatnot.
  6. Nov 30, 2012 #5
    of course there's greub, and I always liked chap 7, & parts of chap 4 of hungerford's algebra. & roman is good too.
    Last edited: Nov 30, 2012
  7. Nov 30, 2012 #6


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    i modestly offer the free notes on my webpage, math 8006a,b,c, which cover up through noetherian modules, jordan forms, and spectral theorems in less than 60 pages,

    as well as the 843-part3 notes on tensor and exterior products, also on that page. but i also agree with the classics recommended above.
  8. Nov 30, 2012 #7
    Thanks everyone for the input. I think I will likely go with Roman's text; although it is admittedly too long to finish over the short winter break, I hope to get through most of Part I and then touch on chapters 11 and 14 (at least).

    Mathwonk - thanks for the reference to your notes, I will likely use them as a study guide as well.
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