Definition of the Tensor Product

Could anyone explain in basic language what the tensor product is? I am relatively new to matrix algebra and I am completely new to this specific concept. What exactly does it do for me and what properties of it should I know?

The tensor product is a gadget which allows you to turn bilinear maps into linear maps between vector spaces. I explain this below.

Let $V,W$ and $X$ be vector spaces. We say a function $f:V \times W \to X$ is bilinear if $f(\alpha v_1 + v_2, w) = \alpha f(v_1,w) + f(v_2,w)$ and $f(v, \beta w_1 + w_2) = \beta f(v, w_1) + f(v, w_2)$ for all $v_1, v_2, v \in V$, $w_1, w_2, w \in W$, $\alpha, \beta \in \mathbb{R}$ (let's work over $\mathbb{R}$). This is just saying that $f$ is linear in each "slot".

Now a tensor product of $V$ and $W$ is a vector space $T$, together with a bilinear map $\otimes$, such that if $f:V \times W \to X$ is bilinear, then there is a unique linear map $\varphi: T \to X$ such that $f = \varphi \circ \otimes$, i.e. $f(x,y) = \varphi(\otimes(x,y)) = \varphi(x \otimes y)$ ($\otimes(x,y)$ is denoted by $x \otimes y$).

Now given vector spaces $V$ and $W$, we can always construct a tensor product. Here is one example:

$\otimes: \mathbb{R}^2 \times \mathbb{R}^2$ given by $x \otimes y = x y^t$ ($y^t$ is the transpose) is a bilinear map. One can check (maybe not so easily) that the vector space $T$, where $T = \operatorname{span}\{x \otimes y | x,y \in \mathbb{R}^2\}$, is a tensor product of $\mathbb{R}^2$ and $\mathbb{R}^2$.

I suggest you pick up a book on multilinear algebra, which will discuss these things. Or read other things on the internet. Dummit and Foote is also good.

 Multilinear Algebra, Werner Greub

 Abstract Algebra, Dummit and Foote

• thank you for your help :) – dreamer Feb 12 '13 at 13:48
• No problem. I'm not sure how much my response helped. But now when I think about it, Abstract Algebra by Dummit and Foote have good explanation of the tensor product. I've added this reference to the post. – nigel Feb 12 '13 at 17:22