Construction of non null section on a linear vector bundle

Question

I think this is quite basic.

Say $E \xrightarrow{\pi} M$ is a linear vector bundle on a smooth variety with positive transition functions, prove that it is a trivial bundle.

In few words I have to find a non null section $\sigma: M \rightarrow E$

Answer 1

I belive that the dimension of the fibres of $E$ is $1$. Let $(U_i)_{i\in I}$ be the trivialization for which the transition functions are positive. Take a partition of unity $f_i$ associated to $(U_i)$, on $U_i$ you have a non trivial section $s_i$ which is strictly positive. Write $s=\sum f_is_i$, it is a non vanishing section.