Let $f: U\rightarrow \mathbb R$ and $g: U\rightarrow \mathbb R^m$ be two $C^1$ functions defined on the open set $U\subset \mathbb R^n$. I'd like to show that if $a\in g^{-1}(\{0\})$ is such that:
- $df_a=0$;
- $dg_a:\mathbb R^n\rightarrow \mathbb R^m$ is surjective;
then there exists a unique linear mapping $\lambda: \mathbb R^m\rightarrow \mathbb R$ such that
$$df_a=\lambda\circ dg_a.$$
The ideia is to show that $\textrm{Ker}(dg_a)\subset \textrm{Ker}(df_a)$ and to show that I must use the implicit function theorem, but I don't really know how to start about that. Any hints or references?