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 :)