Let $M$ be a smooth manifold and fix an open cover $(U_i)_{i \in I}$. Giving a smooth rank-$k$ vector bundle $E \to M$ together with chosen local product charts over the sets $U_i$ is equivalent to giving a smooth $GL(k,\mathbb R)$-valued cocycle $(g_{ij})$ on the cover, in the sense that the two constructions recover one another by a vector bundle isomorphism carrying each chosen local product chart to the corresponding glued chart. Changing the chosen local product charts replaces $(g_{ij})$ by $g'_{ij}(x) = h_i(x)g_{ij}(x)h_j(x)^{-1}$, where $h_i:U_i \to GL(k,\mathbb R)$ are smooth maps.