On Isomorphisms of tensor product I have seen different versions of the isomorphism between linear mappings and the tensor product. The one that I'm currently looking at is this: $\operatorname{Hom}_k(V,W)\cong V^* \otimes W$. My questions are these: is this $\operatorname{Hom}_k(V,W)\cong V \otimes W^*$ true as well? And what happens if we make one of the spaces in the hom dual spaces? E.g. is  $\operatorname{Hom}_k(V^*,W)\cong V \otimes W$ true, i.e. can you just add a star on the "hom-side" and remove a star on the other side corresponding to the same space will still maintaining the isomorphism?
Thanks
Clarification: working in finite dimensional vector spaces

Two finite dimensional vector spaces $V$ and $W$ are isomorphic if and only if their dimensions are the same, since all vector spaces are free.
Hence, your question amounts to calculating the dimension of $\operatorname{Hom}_k(V,W)$, which is $\dim(V)\dim(W)$ and that of $V\otimes W^*,$ which is $\dim(V)\dim(W^*)=\dim(V)\dim(W)$.
