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.
 A: 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.
A: 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://abel.math.harvard.edu/archive/25b_spring_05/tensor.pdf
A: See whether this makes sense
http://stemandmusic.in/maths/mvt-algebra/matrixTP.php
In case it does not, please let me know in comments 
