- #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.
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.