Geometric Hahn-Banach theorem and linear operators

Question

I've got this problem. $(X, ||\cdot||)$ is a normed vectorial space. Then, $l_0, l_1, ..., l_n$ are linear operators in $X^*$ (dual space) and for each $i\in \{0,1,...,n\}$, let $ker\:l_i = \{x\in X | l_i(x) = 0\}$. Let's suppose that: $\bigcap_{i=1}^{n} ker\:l_i \subset ker\:l_0$. And the function $G:X\rightarrow \mathbb{R^{n+1}}: x\rightarrow (l_0(x), l_1(x), ..., l_n(x))$. So now I'm trying to prove that a vector $(\alpha_1, \alpha_2,..., \alpha_n)\in \mathbb{R^n}$ exists, such that $(\forall x\in X)\:l_0(x)=\sum_{i=1}^{n} \alpha_i l_i(x)$. I'm supposed to use Geometric Hahn-Banach but I'm really lost. Any advice or hint as to how to prove this? I've also proven that the vector $(1,0,...,0)\notin G(X)$ so I think I could use that specific vector for the theorem, but I have not been able to get far. Any help would be appreciated.