An exercise from Loring Tu's textbook asks the following question:
Let $\pi:E\to M$ be a $C^\infty$ vector bundle and $s_1,\ldots,s_r$ a $C^\infty$ frame for $E$ over an open set $U$ in $M$. Then every $e\in\pi^{-1}(U)$ can be written uniquely as a linear combination $$ e=\sum_{j=1}^r c^j(e)s_j(p),\quad p=\pi(e)\in U. $$ Prove that $c^j:\pi^{-1}U\to\Bbb R$ is $C^\infty$ for $j=1,\ldots,r$.
The book offers a hint to first show that the coefficients of $e$ relative to the frame $t_1,\ldots,t_r$ of a trivialization are $C^\infty$. I think this follows from an example where it is shown that the frame of a trivialization is $C^\infty$ (Example $12.10$ for those who have the book), but even with this, I'm not sure what to do. Will anyone offer some hints?
