Let $E$ and $M$ be smooth manifolds and let $\pi:E\rightarrow M$ be a smooth surjective map. Suppose that $E$ is a smooth vector bundle of rank $k$ over $M$. Let $\Psi:\pi^{-1}\left(U\right)\rightarrow U\times\mathbb{R}^{k}$ and $\Phi:\pi^{-1}\left(V\right)\rightarrow U\times\mathbb{R}^{k}$ be local trivializations of $E$ over $M$ with $U\cap V\neq\emptyset$ . Then there exists a smooth map $\tau:U\cap V\rightarrow\mbox{GL}\left(k,\mathbb{R}\right)$ such that $\left(\Phi\circ\Psi^{-1}\right)\left(p,v\right)=\left(p,\tau\left(p\right)v\right)$ for all $p\in U\cap V$ , $v\in\mathbb{R}^{k}$.
Here is my attempt of the proof (obtained with help from textbooks):
Let $\pi_{1}:\left(U\cap V\right)\rightarrow\mathbb{R}^{k}$ be the projection onto $U\cap V$ . We have $\pi_{1}\circ\left(\Phi\circ\Psi^{-1}\right)=\pi_{1}$ . This shows that $\left(\Phi\circ\Psi^{-1}\right)\left(p,v\right)=\left(p,\sigma\left(p,v\right)\right)$ for all $p\in U\cap V$ , $v\in\mathbb{R}^{k}$ , where $\sigma:\left(U\cap V\right)\times\mathbb{R}^{k}\rightarrow\mathbb{R}^{k}$ is a smooth map. Moreover, if $p$ is fixed, the map defined by $v\longmapsto\sigma\left(p,v\right)$ is a linear isomorphism. So there exists a function $\tau:U\cap V\rightarrow\mbox{GL}\left(k,\mathbb{R}\right)$ such that $\sigma\left(p,v\right)=\tau\left(p\right)v$ . We need to show that $\tau$ is smooth. Let $e_{1} ,...,e_{k}$ be the standard basis for $\mathbb{R}^{k}$ and let $\varepsilon_{1}, ...,\varepsilon_{k}$ be the dual basis for $\mathbb{R}^{k}$ . Let $\tau_{i,j}\left(p\right)$ be the $\left(i,j\right)$ entry of the matrix $\tau\left(p\right)$ . Then $\tau_{i,j}\left(p\right)=\varepsilon_{i}\left(\tau\left(p\right)e_{j}\right)=\varepsilon_{i}\left(\sigma\left(p,e_{j}\right)\right)$ Since it is a composition of smooth maps, $\tau_{i,j}$ is smooth. Therefore $\tau$ is smooth.
Is the proof correct?