Tensor product of vector space problems

[Gramsci]

Tensor product of vector space problms

Homework Statement

I'm currently reading Halmos's book "Finite dimensional vector spaces" and I find it excellent. However, I'm having some problems with his definition of the tensor product of two vector spaces, and I hope you could help me clear it out. Here's what he writes:
"Definition: The tensor product $$U \otimes V$$ of two finite dimensional vector spaces U and V (over the same field) is the dual of the vector space of all bilinear forms on $$U \oplus V$$ . For each pair of vectors x and y, with x in U and y in V, the tensor product $$z = x*\otimes y$$ of x and y is the element of $$U \otimes V$$ defined by $$z(w) = w(x,y)$$ for every bilinear form w."

Just right before that, he talks about that the definition uses reflexivity to obtain the tensor product of U and V. Where? I'm finding this definition somewhat obstruse. And if you guys have the time, I'd love to see an example of how to really get the tensor product of two vector spaces using this definition.

Say, that I want the tensor product of C^2 and C^3. How?

Homework Equations

The Attempt at a Solution

 

[lippyka]

The definition that Halmos has given is a bit abstract, but it can be explained in simpler terms. Basically, the tensor product of two vector spaces U and V is defined as the space of all linear combinations of elements from U and V that are bilinear in their arguments. That is, for any two vectors x and y, with x in U and y in V, the tensor product z = x*\otimes y is an element of U \otimes V such that z(w) = w(x,y) for every bilinear form w. To illustrate this, let us consider the example you gave of taking the tensor product of C^2 and C^3. First, we need to define the bilinear forms on C^2 and C^3. We can do this by defining a function b:C^2 \times C^3 \rightarrow \mathbb{R} such that b(x,y) = \sum_{i=1}^{2}\sum_{j=1}^{3}a_{ij}x_iy_j, where a_{ij} are some constants. This function is bilinear in its arguments and defines a bilinear form on C^2 and C^3. Next, we can take the dual of this bilinear form to obtain the tensor product of C^2 and C^3. The dual of b is a function b*:C^2 \otimes C^3 \rightarrow \mathbb{R} such that b*(x,y) = b(x,y) for all x in C^2 and y in C^3. This function defines the tensor product of C^2 and C^3. In conclusion, the definition given by Halmos is simply a way of defining the tensor product of two vector spaces in terms of their bilinear forms.