Let V,W be some finite dim. vector spaces and denote the bases of V,W respectively.
Now i want to show that is a basis for the tensor product .
(We have defined the tensor product of vector spaces as a quotient vector space, i hope there is only one such def. such that you are knowing what i'm talking about)
I could show, that this set is a generating system fo . This wasn't so difficult. But i couldn't show that they are linearly independent.
I also try to show that is isomorphic to another vector space with dimesnion n*m, like Hom(V,W). But i have difficulties with this.
Can you give me a hint?