Here is the question.
Assume that $X$ is a smooth tangent vector field of $M$, and $X(p)=0$. Show that there exists finitely several smooth functions $f_i$ and smooth tangent vector fields $X_i$, such that $f_i(p)=0$ and $$ X=\sum\limits_{i}f_iX_i. $$
I have a primary idea: If we consider $X$ in the local coordinate $\{U,\varphi\}$ of $p$, then $X$ can be denoted as $$ X(p)=\sum\limits_{i=1}^na^i(p)\frac{\partial}{\partial x^i}\bigg|_p, $$ where $a^i$ is a smooth function of $M$, $a^i(p)=0$. But this is only a local illustration. Is this idea valid? And how can I find the global version of $X$ on $M$ to solve this problem?
Thanks for sharing your opinions!!