Help understanding tensor products I am struggling to understand tensor products. I will first state what I think I understand and then ask questions. 
Definition of Tensor Product: 
http://en.m.wikipedia.org/wiki/Tensor_product
What I Think I Understand: 
Consider two vector spaces $V$ and $W$. Let $V \times W$ be the Cartesian product of $V$ and $W$.
$F(V\times W)$ is a free vector space (no idea what this is). 
$V \otimes W$ is a vector space. The vectors of this space are defined to be the equivalence classes of $F(V\times W)$ under the following equivalence relations 
\begin{align}
&v, v_1, v_2 \in V; w, w_1, w_2 \in W; c \in K; \\
&(v_1,w) + (v_2,w) \sim (v_1 + v_2,w) \\
&(v,w_1) + (v,w_2) \sim (v,w_1+w_2) \\
&c(v,w) \sim (cv,w) \sim (v,cw)
\end{align}
That's about all I understand. 
My Question:
Is this the right way to approach understanding tensor products? If so, can you either fill in the gaps in my understanding or tell me where can I find more information about them (besides Wikipedia) so I can do so myself? 
 A: The major idea behind the tensor product $V\otimes W$ is that it allows us to study bilinear maps $\omega:V\times W\to Z$ as linear maps $\tilde\omega :V\otimes W\to Z$ on a space like $V\times W$ but with the bilinearity 'built-in', thus reducing the theory of bilinear maps to 'simple' linear algebra. In fact, this construction works for more generally multilinear maps $\omega : V_1\times\dots\times V_n\to Z$ as well ($\tilde\omega : V_1\otimes\dots\otimes V_n\to Z$).
A: Saying something similar to what oldrinb said, the tensor product ( I am assuming here that you are tensoring vector spaces , though a lot of other Mathematical "objects" can be tensored) of two vector spaces is a vector space whose dimension is the product of the dimensions of the respective spaces (and there is a canonical construction of a basis {$v\otimes w$} for $ V\otimes W$) and so that, as oldrinb said, every multilinear map in the product vector space $V \times W$ is uniquely mapped into a linear map defined
in the vector space $V \otimes W$ , i.e., there is a bijection between the set of bilinear maps from $V \times W$ into a third vector space $Z$ (over the same field) and the collection of linear maps from $V \otimes W$  into $Z$.
A: Here's a document, by Keith Conrad, that I found helpful: http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/tensorprod.pdf - I have struggled understanding tensor products, just like you, and I still don't feel completely comfortable wth it. The point of the tensor product is that you can isolate the bilinearity of any bilinear function into a 'canonical' bilinear part and something linear - and because the bilinearity is a standard construction, the tensor product, you can more or less ignore it, because the interesting bits are in the linear "left-overs". 
