Let $M$ be a sub-manifold of $\mathbb{R}^n$. Suppose $M$ contains the origin. Let $T_0M$ be the tangent space on $M$ at $0$ and $F$ is a subspace in $\mathbb{R}^n$ such that $T_0M\bigoplus F=\mathbb{R}^n$. Show that there exists open sets $U\subseteq T_0M, V\subseteq F$ such that $0\in U, 0\in V$ and a smooth map $f:U\to V$ such that $M\cap (U\times V)=graph(f)$.
My idea: when $M=\mathbb{R}^k\times\{0\}^{n-k}$, then for any neighborhoods $U,V$ of $0$ in $\mathbb{R}^k$ and $\mathbb{R}^{n-k}$, define $f:U\to V$ to be the zero function. Clearly, this $f$ satisfies the property that we need to show. What happens for a general sub-manifold $M$?