advice for proving things about the tensor product of vectors?

From: Snis Pilbor (snispilbor_at_yahoo.com)
Date: 03/16/05 

Date: 16 Mar 2005 15:17:52 -0800

Hi,

    I've been through just about every algebra book in the university
library and they're all the same: they have a paragraph, *MAYBE* a
page, about the tensor product of vectors, while having whole chapters
about the tensor product of vector _spaces_, linear transformations
etc. I'm frantic because I have a homework assignment to prove a ton
of little properties about tensor product of vectors; the problems look
simple but there's just nothing to say. You open your mouth to start
saying a proof, and it hits you-- you can't say a thing, because
everything you know is about the tensor product of _spaces_ or
transformations. The canonical concrete construction of the tensor
product of vectors is quite worthless (I'm talking about the quotient
version here, where you include an n-tuple into the free module and add
(take quotient map) a rather complicated mess).
     One book put forth a bunch of little properties about the things,
and the proof went like: "this is immediate". Er, no, it's not
immediate at all! And to make matters worse, the problems are general
in the sense that I'm asked to prove things about tensor product of
many (arbitrary, not necessarily finite) vectors. What little any
texts have to say about the matter at all, is almost entirely
restricted to the case of 2 vectors!
     So if anyone can help - even if just giving references to books
that actually offer insight - it would be greatly valued.