I stuck at the following linear algebra problem. Could you give me some hints?
Let $V$ be a vector space. Given $g,\,f_1,\,f_2,\,...,f_r\in V^*$, prove that $g\in \mathrm{span}\,(f_1,\,f_2,\,...,f_r)$ if and only if $\cap^{r}_{i=1}\mathrm{ker}\,(f_i)\subset\mathrm{ker}\,(g)$.
The "only if" part seems obvious: if $g\in \mathrm{span}\,(f_1,\,f_2,\,...,f_r)$, then there exits scalars $\alpha_1,...,\alpha_r$ such that $g=\Sigma^{r}_{i=1} \alpha_if_i$. Thus for $\forall v\in \cap^{r}_{i=1}\mathrm{ker}\,(f_i)$, we have $g(v)=\Sigma^{r}_{i=1} \alpha_if_i(v)=0$, which implies $v\in \mathrm{ker}\,(g)$.
But for the "if" part, I have trouble showing that if $\cap^{r}_{i=1}\mathrm{ker}\,(f_i)\subset\mathrm{ker}\,(g)$, then $g$ is in the span of $\{f_i\}_{i=1,...,r}$.
(This is a homework problem so hints or the key ideas are preferred. Thank you for your time.)