Let $E \to M$ be a smooth vector bundle and let $T_x : E_x \to E_x$ be a vector space morphism for some $x \in M$.
Can I produce a vector bundle morphism $T : E \to E$ such that $T = T_x$ when restricted to $E_x$?
Clearly if $E = M \times \mathbb R^n$ then we can define $T(m,v) = T(x,v)$ for that fixed $x$. What can we do in general?