TITLE: "Basis extension theorem" for local smooth vector fields
QUESTION [7 upvotes]: Let $\pi: E \to M$  be a smooth vector bundle of rank $n$, and suppose $s_1, \ldots, s_m$ are independent smooth local sections over an open subset $U \subset M$. 
Can I prove the "basis extension theorem" for smooth sections? That is, for each $p \in U$, there are smooth sections $s_{m+1},\ldots,s_n$ defined on some neighborhood of $V$ of $p$ such that $(s_1, \ldots, s_n)$ is a smooth local frame for $E$ over $U \cap V$.

REPLY [6 votes]: Let $W$ be a neighborhood of $p$ in $M$ such that there is a local trivialization $\varPhi:\pi^{-1}(W)\to W\times \mathbb{R}^n$. Then $v_i(q)=(\pi_{\mathbb{R^n}}\circ\varPhi\circ s_i)(q)$ are linearly independent in $\mathbb{R^n}$ for each $q\in W$. We can find $v_m, \ldots,v_n \in \mathbb{R}$ such that $\{v_1(p), \ldots,v_m(p),v_{m+1},\ldots,v_n\}$ is a basis for $\mathbb{R^n}$. So $\det [v_1(p), \ldots,v_m(p),v_{m+1},\ldots,v_n]\neq0$, because the $\det$ is continuous there is a neighborhood $V$ of $p$ such that $\det [v_1(q), \ldots,v_m(q),v_{m+1},\ldots,v_n]\neq0$ for each $q\in V$. Then $(s_1, \ldots s_m,s_{m+1}=\varPhi^{-1}(\cdot,v_{m+1}), \ldots, s_{n}=\varPhi^{-1}(\cdot,v_{n})) $ is a frame over $U\cap V $.