Since we’re looking at vector spaces, which are special kinds of modules, we know that has a tensor product structure. Let’s see what this means when we pick bases.
First off, let’s remember what the tensor product of two vector spaces and is. It’s a new vector space and a bilinear (linear in each of two variables separately) function satisfying a certain universal property. Specifically, if is any bilinear function it must factor uniquely through as . The catch here is that when we say “linear” and “bilinear” we mean that the functions preserve both addition and scalar multiplication. As with any other universal property, such a tensor product will be uniquely defined up to isomorphism.
So let’s take finite-dimensional vector spaces and , and bases of and of . I say that the vector space with basis , and with the bilinear function is a tensor product. Here the expression is just a name for a basis element of the new vector space. Such elements are indexed by the set of pairs , where indexes a basis for and indexes a basis for .
First off, what do I mean by the bilinear function ? Just as for linear functions, we can define bilinear functions by defining them on bases. That is, if we have and , we get the vector
in our new vector space, with coefficients .
So let’s take a bilinear function and define a linear function by setting
We can easily check that does indeed factor as desired, since
so on basis elements. By linearity, they must agree for all pairs . It should also be clear that we can’t define any other way and hope to satisfy this equation, so the factorization is unique.
Thus if we have bases of and of , we immediately get a basis of . As a side note, we immediately see that the dimension of the tensor product of two vector spaces is the product of their dimensions.