Next step in Linear Algebra studies?

  1. Hey folks,
    I just finished reading Axler's Linear Algebra Done Right, and Halmos' Finite Dimensional Vector Spaces, as well as doing pretty much all the problems in both of them. I really like linear algebra, and would like to keep learning more. I am interested in multilinear algebra and tensor algebra. Is there a nice intro to this subject out there, especially one that doesn't rely on abstract algebra (as I don't know much of that)?
    Broccoli 21
  2. jcsd
  3. From what I have seen, advanced linear algebra either requires some abstract algebra (e.g. module theory) or topology (e.g metric spaces). Two books I think seem good are Steven Roman's "Advanced linear algebra" and "The linear algebra a beginning grad student ought to know".
  4. You'll experience some tensors in most analysis on manifolds courses. Check out Calculus on Manifolds by Spivak or Analysis on Manifolds by Munkres. Typical prerequisites are real analysis and rigorous linear algebra.
  5. from there I would say there are two directions you could go. there's functional analysis which is sort of like infinite-dimensional linear algebra or there's modules, which is like linear algebra where the field is replaced by a ring. if you do modules you'd definitely get to tensors at some point, & probably multilinear algebra. before doing any of that though I think it would be a good idea to brush up on abstract algebra, especially rings.
  6. Hmm. Both of these seem to require math that I haven't taken. Would Baby Rudin be sufficient background for functional analysis?
    I have only basic abstract algebra under my belt. Maybe I'll wait till later to tackle modules and the like.
  7. To go any further, your will need to learn Abstract Algebra. As you go along, Linear Algebra and Abstract Algebra merge in the study of modules and algebras. There are a number of solid algebra textbooks, but different ones are better depending on your starting level. Have you done an abstract course, or just read on your own?
    Roman's book is excellent, and a good supplement to a straight algebra book.
  8. Which direction would be most favourable for a physicist to take? I am in a similar position as the OP. Also, could someone recommend a beginners/an introductory book in abstract algebra, i.e. an abstract algebra book for people who have just finished linear algebra? Thanks in advance.
  9. micromass

    micromass 19,216
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    If you want to do functional analysis, then check out Kreyszig. This is a wonderful book that doesn't demand much prereqs. You don't need baby Rudin in order to read that book as it develops analysis from the beginning. (familiarity with a book like Spivaks calculus is necessary though)

    If you want to go the algebra route, then you should read "basic algebra" by Knapp. It's an extremely beautiful and inspiring book.
  10. Algebra wouldn't hurt but functional analysis would probably be better. Kreyszig has a good book which keeps the topology & measure theory to a minimum. I haven't seen a lot of intro algebra books but iirc one of Herstein's is good & has lots of examples.
  11. Thanks! I have taken intro to analysis (using baby Rudin), so I think I'll check out Kreyszig. Functional Analysis seems pretty cool, and like Pattern, I'm interested in Physics.

    Also, after I take abstract algebra I (groups, rings and fields), I can take representation theory. Would that be sufficient background to study Roman's "Advanced linear algebra"? If not, then what other math should I take?

    Note: on the representation theory class description it says:
    "The topics covered will include group rings, characters, orthogonality relations, induced representations, applications of representation theory, and other select topics from module theory."
  12. I think you'll be fine for Roman if you know about groups, rings and fields. He does cover the preliminaries briefly, though.
  13. mathwonk

    mathwonk 9,760
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    Ted Shifrin wrote a book on abstract algebra for people that have just had linear algebra, called algebra from a geometric viewpoint.

    werner greub has two books on linear algebra, one called linear algebra and one called multilinear algebra. I would suggest the second one for the OP's original request.
  14. Thanks! In the description of "Multilinear Algebra", it says that his "Linear Algebra" book is a prerequisite. "Linear Algebra" seems to be a graduate-level book. Should I study that first?
    You can see the table of contents here (look inside)
  15. it's written in a less informal style & from a more abstract point of view than axler's. you could try working on a book on multilinear algebra (northcott does another one btw), say flip through a copy at your library & see if you can follow it. if not I still think more abstract algebra would probably make it easier.
  16. Since you've already studied Halmos' Finite Dimensional Vector Spaces, you might enjoy working the problems in his Linear Algebra Problem Book.
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