Understanding Halmos's Definition of Tensor Product of Vector Spaces

[Gramsci]

Hello,
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.

Hope you have the time.

 

[Landau]

I find this one of the lesser section in the otherwise great book. There are more elegant and insightful definitions of the tensor product. E.g. see http://www.math.harvard.edu/~tomc/math25/tensor.pdf .

 

[Gramsci]

Landau:
Thanks, I'll read it through.
I'd love to see some "concrete" examples too if possible.

 

[Gramsci]

Some notation in there that I'm quite not used to. What's really confusing me are some concrete examples, say, how to use (any given) definitions to form the tensor product of say, C^2 and C^3.

 

[Hurkyl]

An inner product on V is an element of (V (x) V)* satisfying some additional properties.

A linear transformation on V is an element of V (x) V*.

I say "is" loosely. e.g. if you're thinking in terms of linear transformations, then there is a difference between the following:

• A linear transformation V -> W
• A linear transformation on W* -> V*
• An element of W (x) V*
• A bilinear form on W* x V

despite the fact there is a natural, canonical way to convert back and forth between any of them, and upon choosing a basis, they are all naturally represented by matrices.

(In the above, V and W denote finite-dimensional vector spaces)

 

[Gramsci]

Hurkyl:
I'm grateful, but I can't say I understand your point of the message.
I feel really stupid for not understanding this.

 

[Hurkyl]

There wasn't a coherent point, just an information dump!

One thing was to point out that matrices are a familiar example of elements in a tensor product.

Another thing to keep in mind is not to fixate on a particular representation. The "V (x) W consists of bilinear forms on V* x W*" is a common sort of thing, but it's overly restrictive to think in that way.

e.g. just think of all the ways you use a matrix: you can multiply it by a column vector on the right, or a row vector on the left. If you have both a row and a column vector, you can multiply them on either side of the matrix to get a number. If you have another matrix with a compatible dimension, you can multiply them. You can create a matrix by adjoining several compatible matrices side-by-side. You can use a matrix to define a subspace in four different ways: the row space, column space, and its left and right null spaces.

Any tensor product has a similarly bewildering variety of uses. IMHO it's more fruitful to think in terms of "what things can I do with a tensor" rather than trying to fixate on some particular definition as if that's what a tensor "really is".

 

[Fredrik]

This thread might be useful.