What is the isomorphism between E* and Hom(E, M$\times$R)

Question

What is the isomorphism between dual vector bundle $E^*$ and $\mathrm{Hom}(E,M\times \mathbb R)$? There is a natural isomorphism on bundle that is $\mathrm{Hom}(E,E')=E^*\otimes E$, therefore I am wondering if I can use this isomorphism to get the result that $E^*$ is isomorphic to $E^*\otimes (M\times\mathbb R)$ and thus isomorphic to $\mathrm{Hom}(E, M\times \mathbb R)$?

Answer 1

You are right, $E^*$ is isomorphic to $E^*\otimes (M\times\mathbb R)$. This follows from the fact that $V\otimes\mathbb R\cong V$ for every real vector space $V$.