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Good Linear Algebra Text

  1. Oct 18, 2011 #1
    Hey, I was wondering if anyone could suggest a good linear algebra text that gives thorough coverage of vector spaces over all sorts of fields. It's difficult to find a book that doesn't focus pretty much solely on real and complex vector spaces that isn't at too high of a level. (Although I don't want something too introductory. I prefer a feeling of unfamiliarity at first; otherwise I don't feel like I'm gaining anything.)

    Something with a coverage of infinite dimensional vector spaces and and function spaces would be nice also but isn't too necessary.

    Thanks a lot!
     
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  3. Oct 18, 2011 #2

    jk

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    Hoffman and Kunze
     
  4. Oct 18, 2011 #3
    I learned linear algebra from Artin Algebra. Halmos Finite Dimensional Vector Spaces seems like something you may like.
     
  5. Oct 18, 2011 #4

    micromass

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    Hoffman and Kunze is exactly the book you want. It's one of the best books out there. Furthermore it treats the theory in full generality: over any field. Determinants are even treated over arbitrary commutative rings!!
     
  6. Oct 19, 2011 #5
    Awesome, I saw a few references to Hoffman and Kunze as I was searching before I posted this, but not a lot about its coverage of the theory. I'll likely go with that one, but all five of my University's copies are either out or missing, so if it takes too long, Halmos looks like a very good resource too.

    Thanks a lot, everyone!
     
  7. Oct 21, 2011 #6
    "Infinite dimensional" vector spaces and function spaces is what we usually call Functional Analysis, which is usually a graduate-level topic for mathematics students, but you may see it in undergrad courses like "(Advanced) Topics in Analysis". You also need a background in algebra, real and complex analysis, measure theory and maybe topology to study it (basically an undergrad degree), although you can find books with fewer prerequisites, some of them written for physicists. Do you want books on Functional Analysis?
     
    Last edited: Oct 21, 2011
  8. Oct 21, 2011 #7
    I have familiarity with algebra, topology, and real analysis, and and understanding of some of the principles of measure theory, although none of the math itself at the moment. I've not yet been exposed to complex analysis.

    I'm glad you told me all this: I'm an undergrad doing computer science and basically studying pure math of library books in my spare time. This method works very well for me in terms of learning, but the prerequisite chains aren't quite as explicit.

    If you have any recommendations on functional analysis, however, I'd certainly love to write them down for the future. (I know from his foreword that Halmos' text was meant to elucidate the relationship between linear algebra and functional analysis, albeit by studying the finite-dimensional side only.)
     
  9. Oct 21, 2011 #8

    micromass

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    Take Kreyszig's text: introduction to functional analysis. It's one of the best intro texts out there.
     
  10. Oct 21, 2011 #9

    lurflurf

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    Real and complex vector spaces are of particular interest some they get much attention.
    I take it you want a general book without particular focus on computation, applications, or specialized methods.

    Advanced Linear Algebra by Steven Roman is very good, the third edition seem to have fewer errors than the second. It helps to know a little linear algebra going in. It might be what you call "at too high of a level", but it does go beyond finite dimension and real and complex spaces. Also has nice applications and a helpful review of algebra.

    Linear Algebra by Georgie E. Shilov is also good. A pleasure to read.

    Finite-Dimensional Vector Spaces by P.R. Halmos
    Linear Algebra Done Right by Sheldon Axler
    Are good if you read them at just the right time. Halmos keeps to finite dimension, but is written to be good background for infinite dimension later. Axler avoids determinants, focus pretty much solely on real and complex vector spaces and finite dimension, but is good as a first look, or second look if the first was very bad.
     
  11. Oct 23, 2011 #10
    Yes, Kreyszig's book is suitabe for you. It is at a level that undergrad physics students can understand it, and it is written for non-math people, while still being a proper maths book, if that makes any sense. Math students usually take Functional Analysis in the 2nd year of graduate studies, with graduate analysis especially under their belt. Some idea of this can be seen in books like https://www.amazon.com/Course-Funct...922/ref=sr_1_1?ie=UTF8&qid=1319377231&sr=8-1".
     
    Last edited by a moderator: Apr 26, 2017
  12. Oct 24, 2011 #11
  13. Oct 27, 2011 #12
    I think Sheldon Axler's "Linear Algebra Done Right" is a good introduction to what Linear Algebra is truly about- Linear Transformations. It freely uses Real and Complex fields, lumping them under the symbol F (black-board bold). It eschews determinants and matrices, but I think that is a good thing to do if one really wants to deeply understanding Vector Spaces.

    It briefly covers linear functionals, but like others have said, if you want to deal with Infinite-Dimensional Vector Spaces and Linear Functionals, check out a classic text on Functional Analysis.

    Another good book to look at is Peter Lax's "Linear Algebra and Its Applications".
     
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