# Any reference on multilinear algebra [closed]

Do you know any good reference on multilinear algebra?

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## closed as off topic by Harry Gindi, Charles Siegel, Yemon Choi, Scott Morrison♦Mar 9 '10 at 1:31

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At the moment, google is better at answering this question than any of us are. What are your requirements? – Pete L. Clark Mar 8 '10 at 21:32
Closed per Pete's comment. The question isn't specific enough for anyone here to give a more useful answer than google can. – Scott Morrison Mar 9 '10 at 1:31
I'm sorry this question was closed. It had at least one interesting answer. – Ilya Grigoriev Mar 9 '10 at 2:18
I respectfully disagree with Pete's assessment: nothing personal since he left it BEFORE I posted my answer and couldn't have read this answer. Closing the question (which he didn't advocate) will certainly make it a self-fulfilling prophecy... – Georges Elencwajg Mar 9 '10 at 11:43
I have voted to reopen, as I think that mathematicians giving their honest opinion about helpful books is more useful than a random Google search. I personally vote for "Multivariable Mathematics" by Shifrin which presents differential forms at the level of a 1st year undergraduate student. Very concrete (how do you calculate the area of a parallelogram spanned by two vectors in R^4?). – Steven Gubkin Jun 6 '12 at 19:16

Linear Algebra by Hoffman and Kunze covers this in chapter 5, where the tensor and exterior algebras are introduced. Algebra by Serge Lang covers this in more detail in the later chapters, but this is a more difficult and in-depth treatment which also explains the universal properties of the symmetric, exterior, and tensor algebras along with other multilinear constructions.

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Linear Algebra by Hoffman and Kunze is one of my favorite math books! It was my first introduction to multilinear algebra and I highly recommend it. – K.J. Moi Mar 8 '10 at 22:02

Dear mingming, here are three excellent books.

1) Tensor Spaces and Exterior Algebra by Takeo Yokonuma. Translations of Mathematical Monographs, volume 108, AMS 1992

You can browse it in Google books here

2) Laurent Schwartz ( yes, the Fields medalist of distibutions fame) wrote a book, little-known even in France : Les Tenseurs, Hermann, 1998.
It is remarkably well written and contains a wealth of information not found, to my knowledge, in other books. The bad news : it is in French and not translated...

3) Finally there is an amazingly original free book by Sergei Winitzki , Linear Algebra via Exterior Products. Here is the link

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I couldn't easily find 2), but 3) does indeed seem very nice. I still have fear of the exterior product, and this book seems like a great way to mitigate it. It's unfortunate it doesn't seem to get to integration of forms, though. – Ilya Grigoriev Mar 9 '10 at 0:57

For the tensor, exterior and symmetric algebras of a module over a commutative ring I suggest the notes by Murfet http://therisingsea.org/notes/TensorExteriorSymmetric.pdf

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The standard reference is Greub's Multilinear algebra. There's also the book by Northcott.

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BTW, I was surprised that Math Reviews only lists 6 books with "Multilinear algebra" in the title. – lhf Mar 8 '10 at 21:36
I would think that most linear algebra books cover multilinear algebra as well (I haven't read Halmos, but that seems like something he would cover). – Harry Gindi Mar 8 '10 at 21:40
Halmos doesn't really cover multilinear algebra. He has some discussion of the tensor product, but it is too vague to give the reader a feel for what should be going on. So in fact some books on linear algebra aimed at math students stick for the most part to linear algebra. – KConrad Mar 8 '10 at 22:19