# 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

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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The difference between an ordered pair of vectors and a tensor product of two vectors is this:

If you multiply one of the vectors by a scalar and the other by the reciprocal of that scalar, you get a different ordered pair of vectors, but the same tensor product of two vectors.

Similarly with an ordered triple of vectors and a tensor product of three vectors, etc.

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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
I don't have a simple answer to that right now.... Maybe later? –  Michael Hardy May 13 '12 at 22:55
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
Another point on the subtlety of the phenomenon of tensors that are not pure (elementary, monomial, separable, whatever you want to call them) is that they are show up in the mathematical description of entanglement in quantum mechanics. That is, in QM the combined state space for two quantum systems is a (completed) tensor product of the original two spaces, and trying to wrap your head around entangled states of two particles is the conundrum of trying to wrap your head around the idea that some tensors are not pure. –  KCd May 14 '12 at 8:54