Let $E\to M$ and $F\to N$ be smooth vector bundles of ranks $r$ and $s$, let $f:M\to N$ be smooth, and let $\Phi:E\to F$ be a fibrewise [linear map](/page/Linear%20Map) over $f$. Choose local trivializations $E|_U\cong U\times \mathbb R^r$ and $F|_V\cong V\times \mathbb R^s$ with $f(U)\subset V$. Then $\Phi$ is smooth over $U$ iff there is a smooth map $A:U\to \mathbb R^{s\times r}$ such that

\begin{align*} \Phi(x,v)=(f(x),A(x)v) \end{align*}

for all $(x,v)\in U\times \mathbb R^r$.