## Tensor Product

**Point of post:** In this post I will discuss the very basic, and simple minded, definition of the *tensor product * of finite dimensional vector spaces and and it’s consequences, as is outlined in Halmos (viz. reference 1).

*Nota Bene: *The following may seem to be a far-cry from the typical definition of the tensor product as where is the free vector space and is the usual equivalence relation. That said, the following is a fairly large amount of theoretical buck for a fairly small complexity buck.

**Motivation**

In the last post we discussed how given vector spaces over a field there is a canonical way to form the vector space of all bilinear forms on , denoted . But, as is fast becoming a motif in our studies we begin with a vector space and the study it’s dual space, as always denoted either or . For the case of we make a small notational change. Instead of denoting the dual space of by we denote it by and call it the tensor product of and .

What’s the motivation for this space? Why is it important, and more importantly how can we get a *feel *for it. Consider . This is as innocuous of a vector space as one is liable to come across. But, for a second let’s think of it instead of being a set of -tuples, but instead of a set of functions. Namely, let’s define

We can define similarly. Evidently but, really indulging in a logical inaccuracy we can really think of them as being the *same *in a very natural way. That said, we can clearly consider the vector space

Once again, one can easily see that . Clearly then is intimately related to and , but the million dollar question is…how? A quick dimension argument shows that is not isomorphic to . Using another dimension argument we can see that but this doesn’t seem to catch the aforementioned feel we are seeking, this isomorphism doesn’t have a *natural *quality about it. Similarly, what if we considered

and

Then how do these two related to

Both of these give a feel about what the tensor product is. It takes a vector space “acting” on object and one “acting” on objects and produces a vector space “acting” on objects.

*Tensor Product*

Formally, for vector spaces and over the field we define , called the *tensor product *of and , to be

And for and we define the *tensor product *of and by

from where it’s clear that .

*Remark:* From here on it it’s assumed that is a vector space of dimension , a vector space of dimension , both over a field .

**Theorem:** *Let , , and . Then, for any *

*and for any *

*where addition and multiplication are the usual operations with linear functionals.*

**Proof:** Let then

But, since was arbitrary it follows that as required. The case for the second slot is done exactly the same.

*Basis and Dimensionality*

We are now prepared to show that the tensor product’s namesake is well-deserved, namely that it’s a product of some kind. Namely, we will show that admits a very natural basis.

**Theorem:** *Let be a basis for and a basis for . Then, is a basis for .*

**Proof:** Recall that a basis for is where . Thus, we know that this basis admits a dual basis for by . Thus, it suffices to check that . To do this we merely note that

and since a linear functional is determined entirely on the a basis it follows that . The conclusion follows.

**Corollary:**

**References:**

1. Halmos, Paul R. *Finite-dimensional Vector Spaces,*. New York: Springer-Verlag, 1974. Print.

[…] of post: This will be the solutions to the problems in Halmos’s book corresponding to this […]

Pingback by Halmos Section 24 and 25: Tensor Product and Product Bases « Abstract Nonsense | November 1, 2010 |

[…] have previously discussed the notion of the tensor product, but will be considering a slightly different (although […]

Pingback by Representation Theory: The Tensor Product and the Tensor Product of Representations « Abstract Nonsense | February 14, 2011 |