TITLE: Why are tensors a generalization of scalars, vectors, and matrices?
QUESTION [30 upvotes]: Take two vector spaces $V$ and $W$ over a field $F$. One may form the tensor product $V\otimes W$ and it fulfills an universal property. Elements of $V\otimes W$ are called tensors and they are linear combinations of elementary tensors $v\otimes w$, the elementary tensors generate $V\otimes W$.
People from physics think of a tensor as a generalization of scalars, vectors, and matrices, I think and I have seen them tensoring matrices with matrices as entries with matrices and so on.
What does this mean and what has it to do with the definition from above? What "is" a tensor?

REPLY [3 votes]: The following discussion is for finite-dimensional vector spaces only:
The tenor product of two vector spaces $V$ and $W$ arises, because you want to work with bilinear, rather than just linear, maps and functions of $V \times W$. The key observation is that the space of all possible bilinear scalar-valued functions or vector-valued maps is itself a vector space that is spanned by simple bilinear functions constructed by multiplying a linear function on $V$ by a linear function on $W$.
Let $T$ denote the vector space of all scalar-valued bilinear functions on $V \times W$ and observe that there is a natural map $V^* \times W^* \rightarrow T$. Moreover, it is easy to see that $V^* \times W^* $ generates all of $T$ in the sense that its image does not lie in any proper subspace of $T$. So this leads to the notation $T = V^* \otimes W^*$
The next observation is that there is a natural bilinear map $V\times W \rightarrow T^* $ that corresponds to evaluation of the given $(v,w)$ with an element of $T$. Again, the image "spans" all of $T^* $ in the sense that it does not lie in any proper subspace. Moreover, the linear duality between $T$ and $T^* $ corresponds exactly to the evaluation of a bilinear function on an element in $V\times W$. So it is reasonable to denote $T^*$ by $V \otimes W$.
So, morally speaking, $V\otimes W$ is the "smallest" vector space such that there is a bilinear injective map $V \times W \rightarrow V\otimes W$. Of course, this statement can be made precise by defining $V\otimes W$ as a universal object.
Finally, it is not difficult to prove the rather cool (and for me not so obvious) observation that the space of linear maps $V \rightarrow W$ is naturally isomorphic to $W\otimes V^* $. In this sense, tensors generalize the matrices viewed as linear transformations.