Let $E\to X$ be a rank-$k$ smooth vector bundle generated by global sections $s_1,\dots,s_n$. Then the evaluation map
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\begin{align*}
\operatorname{ev}_s:X\times \mathbb R^n \to E
\end{align*}
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is a surjective vector bundle morphism. Its kernel $K:=\ker(\operatorname{ev}_s)$ is a rank-$(n-k)$ smooth vector subbundle of $X\times \mathbb R^n$, and there is a short exact sequence of vector bundles
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\begin{align*}
0 \longrightarrow K \longrightarrow X\times \mathbb R^n \xrightarrow{\operatorname{ev}_s} E \longrightarrow 0.
\end{align*}