Is Hom$(V, W) \simeq V^* \otimes W$ naturally?

Question

When $V, W$ are finite dimensional linear spaces, I think Hom$(V, W) \simeq V^* \otimes W$, but is this isomorphism independent of the choice of basis?

Answer 1

Yes: take the map $V^* \otimes W \rightarrow \mathrm{Hom}(V,W)$ defined as $\alpha \otimes w \mapsto (v \mapsto \alpha(v)w).$