What is the definition of R in the tensor product construction?

In summary, the conversation discusses the construction of the tensor product of two vector spaces and the definition of the quotient space. The definition of R, which is the space generated by three equivalence relations, is also mentioned. The speaker suggests referring to class notes for a more detailed explanation of the topic.
  • #1
amicciulla
1
0
Hello,

So I'm trying to understand the construction of the tensor product of 2 vector spaces as stated in the http://en.wikipedia.org/wiki/Tensor_product" . Now, in the article it states that the tensor product of two vector spaces V and W is the quotient space F( VxW )/R (F( VxW ) being the free vector space over VxW). I'm slightly confused about the definition of R, which is defined as the space generated by the 3 following equivalence relations: (v+u,w) ~ (v,w)+(u,w), (v,u+w) ~ (v,u)+(v,w), and k*(v,w) ~ (k*v,w) ~ (v,k*w). Could anybody elaborate on this? How does one generate a space from equivalence relations?

-Adam
 
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  • #2
you subtract the things you want to be equivalent and set those differences equal to zero. then take the space those differences generate.

if you go to my website and open up the class notes for 845-3, on page 23-28 you will find a complete discussion, and a precisely correct one.
 

Related to What is the definition of R in the tensor product construction?

1. What is the Tensor Product Construction?

The Tensor Product Construction is a mathematical operation that combines two vector spaces to form a new vector space. It is often used in linear algebra and is denoted by the symbol ⊗. It allows for the creation of a larger vector space from two smaller vector spaces.

2. How is the Tensor Product Construction calculated?

The Tensor Product Construction is calculated by taking the outer product of two vectors. This involves multiplying each element of one vector by each element of the other vector and then arranging the resulting products into a new matrix. This matrix represents the new vector space created by the Tensor Product Construction.

3. What are the properties of the Tensor Product Construction?

The Tensor Product Construction has several important properties. It is associative, meaning that the order of operations does not matter. It is also distributive, meaning that it follows the distributive law of multiplication. Additionally, it is bilinear, meaning that it is linear in each of its arguments.

4. What is the significance of the Tensor Product Construction?

The Tensor Product Construction has many important applications in mathematics and physics. It is used in quantum mechanics to describe the state space of composite systems. It is also used in multilinear algebra to define multilinear maps. Additionally, it has applications in computer science and engineering, particularly in the field of signal processing.

5. Are there any limitations to the Tensor Product Construction?

One limitation of the Tensor Product Construction is that it can only be applied to vector spaces. It cannot be used with other mathematical structures, such as groups or rings. Additionally, the Tensor Product Construction can become very complex and difficult to calculate when dealing with higher dimensional vector spaces.

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