Is there a more intuitive way to define tensor products other than using free vector spaces? Tensor products come up a lot in some literature I am reading. but every time I go to Wikipedia, it says a prerequisite for understanding tensor products is understanding free vector spaces. Having read the Wikipedia page,  I still don't understand what a free vector space is. I get regular ones but not this "free" kind. so I then cannot understand tensor products. 
So is there a way to explain tensor products in a more intuitive way than using free vector spaces? 
 A: Basically, the tensor product of 2 vector spaces E and F over a field K (or 2 modules over a commutative field) is a vector space G which is a solution to the following so-called `universal problem:
1) There exists a bilinear map $\varphi\colon E\times F\longrightarrow G$.
2) For any vector space L and any bilinear map $f\colon E\times F\longrightarrow L$, there exists a unique linearmap $u\colon G \longrightarrow L$ such that $f=u\circ \varphi$ (this is best described as a commutative diagram).
Technically a solution to this problem is obtained with the vector space $ K^{(S)}= $ families of elements of K indexed by S with finite support. In case S is finite, you can think of, say $K^3$, whose elements are all triples $(x_1, x_2, x_3)$ of elements of K.
A: Other opinion: To start to grasp the concept of tensor or tensors in general on a secure footing you have to understand vector duality and how the tensor product of rank-one-tensors to build the rank-two-tensors which are also dubbed bilinear forms or bilinear maps.
