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Advanced Linear Algebra

  1. Oct 4, 2007 #1
    I am currently taking Advanced Linear Algebra the course description is as follows:

    A Mat 424 Advanced Linear Algebra (3)

    Duality, quadratic forms, inner product spaces, and similarity theory of linear transformations. Prerequisite(s): A Mat 220

    Can any one recommend a good book and/or website to learn this stuff? My professor just lectures and it makes it very difficult to do the HW or really fully understand any thing based on just my notes.

    Thank you!
     
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  3. Oct 4, 2007 #2

    mathwonk

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    i have explained all this in my algebra course notes on my webpage. both in a very short version, 15 pages total, and a very long version, over 300 pages, math 843-4-5, and an in between version, which is still quite dense, under 100 pages,

    or you could spend around 50 bucks and buy the best algebra book available for undergrads, namely algebra by mike artin, and you will have the polished version of the course he taught for years at mit.
     
  4. Oct 5, 2007 #3
    Although I am a bit confused by the format of your response, I thank you for your help!
     
  5. Oct 5, 2007 #4

    morphism

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    You can also try to find a copy of Finite-Dimensional Vector Spaces by Halmos.
     
  6. Oct 5, 2007 #5
    If you want something that goes over the basics in a very understandable way, I prefer the Linear Algebra by Friedburg, Insel and Spence (5th edition, I think). It's not a text that will go into very advanced theory (if you're looking for a graduate text) but for a junior or senior level class it's very good. It's also good because they begin the text with the theory of vector spaces (instead of matrices) which, I think, gives you a fuller understanding of the linear transformation for its properties instead of learning that "transformations are matrices" (this is handy for computing things, but it's a terrible understanding of the depth of the subject).
     
  7. Oct 5, 2007 #6

    morphism

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    Friedberg is suitable for a first course really. For example, dual spaces are only mentioned in passing, and quotient spaces are only mentioned in the exercises. What I gathered from the OP is that he/she is looking for a "high brow" treatment of linear algebra, and Halmos is good for that. Maybe also Hoffman and Kunze; I remember liking their treatment of direct sum decompositions, although it took me a few rounds of reading to understand the big picture. And if the OP has some experience with abstract algebra, then in addition to mathwonk's recommendation of Artin (although I don't remember if Artin talks about dual spaces), I recommend Herstein (who does).
     
  8. Oct 5, 2007 #7

    mathwonk

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    has anyone ever read my 15 page or even 100 page treatment? these are at least as "high brow" as halmos, probably much more so.

    actually the linear algebra section of my 100 page algebra notes is only 21 pages long. it seems i am just not capable of wasting more than about 20 pages on such a trivial subject.

    give it a try and see what you think.


    halmos was a very interesting man, but he wrote back in the 60's when many math books did not really try to explain anything as i recall. hopefully his finite dimensional vector spaces is an exception, but i would be careful. certainly i never learned anything from his measure theory book, except answers to stupid exam questions.

    i benefited from him as a person though; he was a stimulating curmudgeon. when he challenged me as to why i did not have a phd, i was motivated to go and get one, and i did. i am grateful to him for that. when i said i was content to just do good mathematics without a phd his exact words were: "that's a cop - out".

    some people have the ability to piss me off in a stimulating way like that.
     
    Last edited: Oct 5, 2007
  9. Oct 6, 2007 #8

    Chris Hillman

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    Mathwonk was recommending an excellent and widely used algebra textbook: Michael Artin, Algebra, Prentice Hall, 1991. MA is a leading mathematician; his father Emil is the father of "Galois-theory-as-you-know-it".

    As for the book by Paul R. Halmos, Finite-Dimensional Vector Spaces, Springer, 1974, that is the book from which I taught myself linear algebra (by reading, not a course). Needless to say I highly recommend it--- I think it is a brilliant and very clear book. Halmos was not only a legendary expositor of mathematics (his style is both memorable and inimitable) but also a leading researcher, remembered as a father of functional analysis and to the spectral approach to ergodic theory. The interesting thing is that these topics involve infinite dimensional vector spaces; Halmos says in the preface of his textbook that he wanted to prepare students to learn what parts of the finite dimensional theory generalize to infinite dimensions (with appropriate elaboration, of course).

    But if you like the look of the textbook by Artin, well, that is an excellent book also. There are a large number of other books on "abstract" linear algebra, but these are the two which come to mind first. To be precise: I might recommend that you read the appropriate portions of the classic by Birkhoff and Mac Lane, Modern Algebra (which also covers groups, rings, and modules) and that you finish your reading by studying Herstein, Topics in Algebra. Some others here feel that the latter book is not a good first book, but it was the textbook used in the modern algebra course I took as an undergraduate at Cornell, and again I feel it has much to recommend it. Just recently we had cause to quote it in discussing the generalization to [itex]SO(n)[/itex] of the familiar fact that rotations in [itex]SO(3)[/itex] are specified by giving an axis (two real numbers) and an angle (one real number); the generalization is terribly imporant but not even mentioned in other textbooks! See Terry Tao's remark in http://terrytao.wordpress.com/2007/09/29/ratners-theorems/ on a one-dimensional subgroup which is dense in the two-dimensional abelian subgroup of [itex]SO(4)[/itex] and which is not a circle--- this is actually an illustration of something cool in ergodic theory; see above!--- and see John Baez's occasional discussions of maximal tori in compact simple Lie groups.
     
    Last edited: Oct 6, 2007
  10. Oct 7, 2007 #9

    mathwonk

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    I will be interested to learn whether those classic 60's books you recommend are still readable by todays undergrads Chris. Perhaps they are by the ones who visit here.

    of course i still recommend courant, from my undergraduate days.
     
  11. Oct 7, 2007 #10

    Chris Hillman

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