Annihilator of intersection of kernel of independent functionals In the proof of corollary 3.4 in this handout written by Brian Conrad, appears the following:

Since the $\mathrm df_j(z)$'s are linearly independent functionals, the intersection $\mathrm T_z(Z)$ of their kernels is annihilated by exactly those functionals on $\mathrm T_z(X)$ that are linear combinations of the $\mathrm df_j(z)$'s in $\mathrm T_z (X)^\vee$. (This is a general fact in linear algebra that is easily proved by extending the independent functionals to a basis of the dual space, and then computing in a dual basis of the original space.)

If I understand correctly, the general fact is:
Claim. Suppose $\ell_1,\dots ,\ell _n$ are linearly independent functionals. Then $\mathrm{Ann}(\bigcap_i \operatorname{Ker}\ell_i)=\operatorname{Span} \left\{ \ell_i \right\}_i $.
I can see that $\supset$ clearly holds, but I don't see how the independence can be used to prove the converse inclusion. Did I understand the claim correctly? How to prove it?
 A: Let $V$ be an $n$-dimensional vector space over $\mathbb{F}$ and let $\ell_1, \dots, \ell_k$ be linearly independent elements of $V^{*}$. Then:
$$ \operatorname{Ann} \left( \cap_{i=1}^k \ker \ell_i \right) = \operatorname{Span} \{ \ell_1, \dots, \ell_k \}. $$
To prove this, complete $(\ell_i)_{i=1}^k$ to a basis $(\ell_i)_{i=1}^n$ of $V^{*}$ in an arbitrary way (here we use the linear independence) and let $v_1, \dots, v_n$ be the corresponding dual basis of $V$ (so that $\ell_i(v_j) = 0$ for $i \neq j$ and $\ell_i(v_i) = 1$ for $1 \leq i \leq n$). Let $\varphi \in \operatorname{Ann} \left( \cap_{i=1}^k \ker \ell_i \right)$ and write it as $\varphi = \sum_{i=1}^n c_i \ell_i$ for some $c_i \in \mathbb{F}$. If $j > k$ then $v_j \in \ker \ell_i$ for $1 \leq i \leq k$ which shows that $v_j \in \cap_{i=1}^k \ker(\ell_i)$ and so
$$ 0 = \varphi(v_j) = \sum_{i=1}^n c_i \ell_i(v_j) = c_j. $$
Hence, we have shown that $c_i = 0$ for $k + 1 \leq i \leq n$ and hence $\varphi \in \operatorname{Span} \{ \ell_1, \dots, \ell_k \}$.
