Let $\pi:E\to M$ be a smooth real vector bundle of rank $k$ over a smooth manifold $M$. Let $\{U_i\}_{i\in I}$ be an open cover of $M$ with local trivializations
for all $x\in U_i\cap U_j\cap U_k$. Consequently, they define a smooth vector bundle $E_\rho\to M$ with fibre $V$. Moreover, changing the local trivializations of $E$ changes the cocycle $(h_{ij})$ by a smooth coboundary, so the isomorphism class of $E_\rho$ depends only on the isomorphism class of $E$ and on the representation $\rho$.