The Unapologetic Mathematician

Mathematics for the interested outsider

Tensor Products and Bases

Since we’re looking at vector spaces, which are special kinds of modules, we know that \mathbf{Vec}(\mathbb{F}) 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 U and V is. It’s a new vector space U\otimes V and a bilinear (linear in each of two variables separately) function B:U\times V\rightarrow U\otimes V satisfying a certain universal property. Specifically, if F:U\times V\rightarrow W is any bilinear function it must factor uniquely through B as F=\bar{F}\circ B. 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 U and V, and bases \left\{e_i\right\} of U and \left\{f_j\right\} of V. I say that the vector space with basis \left\{e_i\otimes f_j\right\}, and with the bilinear function B(e_i,f_j)=e_i\otimes f_j is a tensor product. Here the expression e_i\otimes f_j is just a name for a basis element of the new vector space. Such elements are indexed by the set of pairs (i,j), where i indexes a basis for U and j indexes a basis for V.

First off, what do I mean by the bilinear function B(e_i,f_j)=e_i\otimes f_j? Just as for linear functions, we can define bilinear functions by defining them on bases. That is, if we have u=u^ie_i\in U and v=v^jf_j\in V, we get the vector

B(u,v)=B(u^ie_i,v^je_j)=u^iv^jB(e_i,f_j)=u^iv^je_i\otimes f_j

in our new vector space, with coefficients u^iv^j.

So let’s take a bilinear function F and define a linear function \bar{F} by setting

\bar{F}(e_i\otimes f_j)=F(e_i,f_j)

We can easily check that F does indeed factor as desired, since

\bar{F}(B(e_i,f_j))=\bar{F}(e_i\otimes f_j)=F(e_i,f_j)

so F=\bar{F}\circ B on basis elements. By linearity, they must agree for all pairs (u,v). It should also be clear that we can’t define \bar{F} any other way and hope to satisfy this equation, so the factorization is unique.

Thus if we have bases \left\{e_i\right\} of U and \left\{f_j\right\} of V, we immediately get a basis \left\{e_i\otimes f_j\right\} of U\otimes V. As a side note, we immediately see that the dimension of the tensor product of two vector spaces is the product of their dimensions.

May 23, 2008 Posted by | Algebra, Linear Algebra | 5 Comments