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.

Solution 1:

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

Solution 2:

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.