# What exactly is a tensor product?

This is a beginner's question on what exactly is a tensor product, in laymen's term, for a beginner who has just learned basic group theory and basic ring theory.

I do understand from wikipedia that in some cases, the tensor product is an outer product, which takes two vectors, say $\textbf{u}$ and $\textbf{v}$, and outputs a matrix $\textbf{uv}^T$. ($\textbf{u}$ being a $m\times 1$ column vector and $\textbf{v}$ being a $n\times 1$ column vector)

How about more general cases of tensor products, e.g. in the context of quantum groups?

Sincere thanks.

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What kinds of objects are you taking the tensor product of? –  Alex Becker May 13 '12 at 7:21
Look at this dpmms.cam.ac.uk/~wtg10/tensors3.html –  Ehsan M. Kermani May 13 '12 at 7:24
+1 for ehsanmo's fantastic link. The explanations there are due to Timothy Gowers, a Fields medalist and one of the outstanding mathematicians of this century (the linked page modestly does not mention the author !) –  Georges Elencwajg May 13 '12 at 7:48
I would add to the list of references two expository articles on the tensor product by Keith Conrad. The link to the web page containing all his expository stuff is this:math.uconn.edu/~kconrad/blurbs. They are a great resource for algebra. The relevant articles are under the section titled Linear/ Multilinear Algebra. –  Rankeya May 13 '12 at 15:06
As a believer in simplicity, I've posted what may be the simplest P.O.V. on this question below. Definitely my answer is not the last word though---i.e. there's lots more to know. –  Michael Hardy May 13 '12 at 17:50
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If you want to study a mathematical object, whether it is a set, manifold, group, vector space, whatever, it is often fruitful to look at natural collections of functions on that space.

Roughly, the purpose of the tensor product, $\otimes$, is to make the following statement true: $$\text{functions}(X \times Y) = \text{functions}(X)\otimes \text{functions}(Y)$$

The specific details about which spaces of functions to choose depend on the type of mathematical object you are interested in.

Here's a pdf that explains it better than I can, http://www.math.harvard.edu/archive/25b_spring_05/tensor.pdf

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I've thought along these lines before (tensors are combos of vectors with special rules), but always ran into the same intuitive stumbling block. Why is it that some elements of a tensor space are pure and can be represented as $w=x \otimes y$, whereas others are composite and can only be built from sums of pure tensors? I sort of gave up on thinking of it that way, but maybe you have some insight..? –  Nick Alger May 13 '12 at 21:36
There is a simple answer: just think about multivariable polynomials! Let's just look at polynomials in two variables. Some of them are monomials, or more generally products $f(x)g(y)$, but most multivariable polynomials are not of that (separable) form. The theory of polynomials in two variables is more than the theory of things that factor as $f(x)g(y)$. This in fact is a special instance of tensor products, since $F[x] \otimes_F F[y] \cong F[x,y]$ for any field $F$. –  KCd May 14 '12 at 8:51