Let $E\to M$ be a smooth vector bundle, let $k\in\mathbb{N}\cup\{0\}$, and suppose $S\subset E$ is a subset such that each $S_p=S\cap E_p$ is a vector subspace of $E_p$. Then $S$ is the total space of a rank $k$ smooth vector subbundle of $E$ iff for every $p\in M$ there is an open neighbourhood $U\subset M$ of $p$ and smooth local sections $s_1,\dots,s_k\in \Gamma(U,E)$ such that

\begin{align*} S_q=\operatorname{span}\{s_1(q),\dots,s_k(q)\} \end{align*}

for all $q\in U$, and $s_1(q),\dots,s_k(q)$ are linearly independent for all $q\in U$.