Tensor product vector spaces over complex and real

[ihggin]

Let U and V be vector spaces over the complex numbers C. Then the tensor product over C, U\otimes_CV is also a complex vector space. Note that U, V, and U\otimes_CV can be regarded as vector spaces over the real numbers R as well. Also note that we can form U\otimes_RV. Question: are U\otimes_CV and U\otimes_RV isomorphic as real vector spaces?

Using the easiest example I could think of, I tried taking U=V=C. Then we have C\otimes_CC\approx C. Since the dimension of C over R is 2, we have that the dimension of C\otimes_CC over R is 2 as well. Next I tried getting the dimension of C\otimes_RC 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.

 

[micromass]

If U and V are vector spaces, then dim(U\otimes V)=dim(U)dim(V). I think this could be useful...

 

[mathwonk]

equivalently, try writing down bases.

 

[ihggin]

Okay, thanks for the tips. Either way, I get \dim_R C\otimes_RC = 4.

 

[Landau]

That is correct :)