Tensor product vector spaces over complex and real

In summary, the tensor product of two vector spaces over the complex numbers is also a complex vector space. However, it can also be regarded as a vector space over the real numbers. The question is whether the tensor products over C and R are isomorphic as real vector spaces. By taking U=V=C, it can be shown that C\otimes_CC and C\otimes_RC have different dimensions, indicating that they are not isomorphic. The dimension of C\otimes_CC over R is 2 and the dimension of C\otimes_RC over R is 4. This can be proven using the fact that dim(U\otimes V)=dim(U)dim(V) and by considering bases.
  • #1
ihggin
14
0
Let U and V be vector spaces over the complex numbers C. Then the tensor product over C, [tex]U\otimes_CV[/tex] is also a complex vector space. Note that U, V, and [tex]U\otimes_CV[/tex] can be regarded as vector spaces over the real numbers R as well. Also note that we can form [tex]U\otimes_RV[/tex]. Question: are [tex]U\otimes_CV[/tex] and [tex]U\otimes_RV[/tex] isomorphic as real vector spaces?

Using the easiest example I could think of, I tried taking U=V=C. Then we have [tex]C\otimes_CC\approx C[/tex]. Since the dimension of C over R is 2, we have that the dimension of [tex]C\otimes_CC[/tex] over R is 2 as well. Next I tried getting the dimension of [tex]C\otimes_RC[/tex] over R, but I couldn't figure it out. My strategy is to show the dimensions are not the same to prove that the two spaces are not isomorphic as real vector spaces.
 
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  • #2
If U and V are vector spaces, then [tex]dim(U\otimes V)=dim(U)dim(V)[/tex]. I think this could be useful...
 
  • #3
equivalently, try writing down bases.
 
  • #4
Okay, thanks for the tips. Either way, I get [tex]\dim_R C\otimes_RC = 4[/tex].
 
  • #5
That is correct :)
 

What is a tensor product vector space over complex and real?

A tensor product vector space over complex and real is a mathematical concept in linear algebra and functional analysis where two vector spaces are multiplied together to form a new vector space. It is a way to combine the elements of two vector spaces to create a new set of elements that have both properties of the original spaces.

What are the properties of a tensor product vector space?

The properties of a tensor product vector space include bilinearity, associativity, and distributivity. Bilinearity means that the tensor product is linear in both of its arguments, associativity means that the order of multiplication does not matter, and distributivity means that the tensor product distributes over vector addition.

How is a tensor product vector space represented mathematically?

In mathematics, a tensor product vector space is represented using the symbol ⊗, which is read as "tensor product." For example, the tensor product of two vector spaces V and W would be written as V ⊗ W. The elements of the tensor product are written as linear combinations of the basis elements of the two original vector spaces.

What is the difference between a tensor product vector space over complex and real?

The main difference between a tensor product vector space over complex and real is the underlying field used in the vector spaces. A complex tensor product vector space is formed by multiplying two complex vector spaces, while a real tensor product vector space is formed by multiplying two real vector spaces. This difference affects the properties and structure of the resulting tensor product vector space.

What are the applications of tensor product vector spaces over complex and real?

Tensor product vector spaces have various applications in mathematics, physics, and engineering. They are used in quantum mechanics, general relativity, and computer graphics to model and analyze physical systems. They also have applications in signal processing, machine learning, and data analysis.

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