Given vector bundles $E \subseteq F$ of rank $r$ and $n$ respectively. I have to show that there exists local trivialisations $U_i$ such that transition maps $E_{ij}$ and $F_{ij}$ satisfies-
$$ F_{ij} = \begin{bmatrix} E_{ij} & * \\ 0 &* \end{bmatrix} $$ I guess it is enough to show that there exists local trivialisations of $F$, $ \varphi_i:U_i \times \Bbb R^n \to \pi_F^{-1}(U_i)$ such that $\varphi_i|U_i \times \Bbb R^r$ is a trivialisation for $\pi_E^{-1}(U_i)$. This is equivalent to the existence of linearly independent sections ${X_i}$ over $U_i$ such that $X_1, … X_r$ belongs to $\pi_E^{-1}(U_i)$. $i.e$ extending a basis $X_1(u), … X_r(u)$ for $E_u$ to $F_u$.
Is my reasoning correct? If it is, I am able to show such a continuous, smooth or holomorphic extension is possible and we’re done.