Let \(p \colon E \to B\) be a vector bundle of rank \(k\text{.}\) Suppose that \(\Phi \colon p^{-1}(U) \to U \times \mathbb{R}^k\) and \(\Psi \colon p^{-1}(V) \to V \times \mathbb{R}^k\) are two local trivializations with \(U \cap V \neq \emptyset\text{.}\) Show that there is a continuous map \(\tau \colon U \cap V \to \mathrm{GL}(k, \mathbb{R})\text{,}\) where \(\mathrm{GL}(k, \mathbb{R})\) is the space of invertible real \(k \times k\) matrices, such that the composite
\begin{equation*}
\Phi \circ \Psi^{-1} \colon (U \cap V) \times \mathbb{R}^k \longrightarrow (U \cap V) \times \mathbb{R}^k
\end{equation*}
has the form \((\Phi \circ \Psi^{-1})(p, v) = (p, \tau(p)v)\text{.}\)
