Tensor product of vector space problems

In summary, the tensor product of two vector spaces U and V is defined as the dual of the vector space of all bilinear forms on U \oplus V. This can be illustrated with an example of taking the tensor product of C^2 and C^3, where the dual of the bilinear form on C^2 and C^3 defines the tensor product.
  • #1
Gramsci
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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 [tex]U \otimes V [/tex] 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 [tex] U \oplus V [/tex] . For each pair of vectors x and y, with x in U and y in V, the tensor product [tex] z = x*\otimes y [/tex] of x and y is the element of [tex] U \otimes V [/tex] defined by [tex] z(w) = w(x,y) [/tex] 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?


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The Attempt at a Solution

 
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  • #2
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.
 

FAQ: Tensor product of vector space problems

1. What is a tensor product of vector spaces?

A tensor product of vector spaces is a mathematical operation that combines two vector spaces to create a new vector space. It is used to represent the relationship between different vector spaces and is an important tool in linear algebra and multilinear algebra.

2. How do you calculate the tensor product of vector spaces?

The tensor product of two vector spaces, V and W, is denoted as V ⊗ W. To calculate it, you take the Cartesian product of the two vector spaces, and then you quotient this product by a certain subspace that represents the bilinearity of the tensor product. The resulting quotient space is the tensor product of V and W.

3. What is the significance of tensor products in physics?

Tensor products are heavily used in physics to describe the relationships between different physical quantities and their transformations. They are particularly useful in the fields of relativity, quantum mechanics, and electromagnetism. Tensor products also help to simplify complex equations and make them more manageable for calculations.

4. Can you give an example of a tensor product of vector spaces?

One example of a tensor product of vector spaces is the cross product of two 3-dimensional Euclidean vector spaces. The cross product combines two vectors to create a new vector that is perpendicular to both of the original vectors. This operation is commonly used in physics to calculate the torque of a force acting on an object.

5. What are some applications of tensor product of vector spaces?

Tensor products have various applications in mathematics, physics, and engineering. They are used in quantum mechanics to describe the states of multiple particles, in differential geometry to represent the curvature of space, and in computer graphics to model the deformation of objects. They are also used in machine learning algorithms for data processing and feature extraction.

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