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Let $V$ be a finite-dimensional real vector space of dimension $k$ and $$f_1,\ldots,f_r:\mathbb{R}^n\to V$$ smooth functions such that $\{f_1(x),\ldots,f_r(x)\}$ is linearly independent for all $x\in\mathbb{R}^n$.

Can we always find $f_{r+1},\ldots,f_k:\mathbb{R}^n\to V$ such that $\{f_1(x),\ldots,f_k(x)\}$ is a basis for $V$ for all $x\in \mathbb{R}^n$?

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    Is $V$ also a smooth manifold? Otherwise the notion of a smooth map $\mathbb R^n\to V$ is undefined. – 2017-01-04
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    @MichaelM Yes, it is a finite dimensional real vector space. – 2017-01-04

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