Exercise 11.10 (The Dual Bundle)

Suppose $M$ is a smooth manifold and $E\to M$ is a smooth vector bundle over $M$. Define the dual bundle to $E$ to be the bundle $E^{\text{∗}}\to M$ whose total space is the disjoint union $E^{\text{∗}}={\coprod <!--\; FUNCTION\; APPLICATION\; -->}_{p\in M}{E}_p^{\text{∗}}$, where $E_p^{\text{∗}}$ is the dual space to $E_p$, with the obvious projection. Show that $E^{\text{∗}}\to M$ is a smooth vector bundle, whose transition functions are given by ${\tau }^{\text{∗}}(p)={\left(\tau {(p)}^{-1}\right)}^T$ for any transition function $\tau :U\to \mathrm{GL}(k,ℝ)$ of $E$.