Tensor Product of C with itself over R.

In summary, the conversation is about proving that C\otimesC (taken over R) is equal to C^2 using the method of complexification. The method involves showing several equivalences, but the speaker is having trouble with one part where they are trying to use the universal property of the tensor product. They have defined a bilinear map and a linear map, but are unable to find an inverse. They are seeking help and resources on complexification for any real vector space.
  • #1
Monobrow
10
0
I am trying to prove that C[itex]\otimes[/itex]C (taken over R) is equal to C^2. The method I have seen is to show the following equivalences:
C[itex]\otimes[/itex]C = C[itex]\otimes[/itex](R[T]/<T^2+1>) = C[T]/<T^2+1> = C.
(All tensor products taken over R).

The only part I am having trouble with is showing that C[itex]\otimes[/itex](R[T]/<T^2+1>) = C[T]/<T^2+1>. I have tried to show this by using the universal property of the tensor product. First I defined a bilinear map from CXR[T]/<T^2+1> to C[T]/<T^2+1> by sending an element (z,f(T)) to zf(t) (where I am omitting the cosets). This then induces a linear map from the tensor product. However, I cannot seem to find an inverse for this map.

Any help would be greatly appreciated.
 
Physics news on Phys.org

What is the definition of Tensor Product of C with itself over R?

The Tensor Product of C with itself over R is a mathematical operation that combines two vector spaces, C and R, to create a new vector space. It is denoted as C⊗R and is defined as the set of all possible linear combinations of elements from C and R with certain properties.

How is the Tensor Product of C with itself over R calculated?

To calculate the Tensor Product of C with itself over R, we use the following formula: C⊗R = {(c,r) | c ∈ C, r ∈ R}. This means that the elements of the new vector space are pairs of elements from C and R, and the operations of addition and scalar multiplication are defined in a specific way.

What are the properties of Tensor Product of C with itself over R?

The Tensor Product of C with itself over R has several important properties, including bilinearity, associativity, and commutativity. This means that the operation is linear with respect to both vector spaces, the order in which the vector spaces are combined does not matter, and the result is the same regardless of the order.

What are the applications of Tensor Product of C with itself over R?

The Tensor Product of C with itself over R has many applications in mathematics, physics, and engineering. It is used in the study of multilinear algebra, quantum mechanics, and signal processing. It is also used in the construction of tensor fields, which are important in differential geometry and general relativity.

What are some examples of Tensor Product of C with itself over R?

One example of Tensor Product of C with itself over R is the construction of the complex numbers, which can be thought of as the Tensor Product of R with itself. Another example is the construction of the quaternions, which can be thought of as the Tensor Product of C with itself. In general, any two vector spaces can be combined using the Tensor Product operation.

Similar threads

  • Linear and Abstract Algebra
Replies
10
Views
197
  • Linear and Abstract Algebra
Replies
2
Views
848
  • Linear and Abstract Algebra
Replies
32
Views
3K
  • Linear and Abstract Algebra
Replies
13
Views
855
  • Linear and Abstract Algebra
Replies
1
Views
877
  • Linear and Abstract Algebra
Replies
2
Views
869
  • Linear and Abstract Algebra
Replies
5
Views
1K
  • Linear and Abstract Algebra
Replies
1
Views
788
  • Linear and Abstract Algebra
Replies
1
Views
780
Replies
4
Views
952
Back
Top