Why is subspace $\mathcal{C}$ the intersection of the kernels of $n-d$ linear forms? I was reading Waldschmidt's notes on Finite fields and error coding, where I came across the following statement, in section $\S 3.3$:

A subspace $\mathcal{C}$ of $F_q^n$ of dimension $d$ can be described by giving a basis ${e_1, . . . , e_d}$ of $\mathcal{C}$ over $F_q$, so that $\mathcal{C} = \{m_1e_1 + · · · + m_de_d | (m_1, . . . , m_d) \in F_q^d \}$. An alternative description of a subspace $\mathcal{C}$ of $F_q^n$ of codimension $n−d$ is by giving $n−d$ linearly independent linear forms $L_1, . . . , L_{n−d}$ in n variables $x = (x_1, . . . ,x_n)$ with coefficients in $F_q$, such that $$(*)\quad\mathcal{C} = \ker L_1 \cap · · · \cap \ker L_{n−d}.$$

I am aware that a subspace is always the kernel of a linear map and vice versa. However, I don't see how $\mathcal{C}$ can alternatively be represented as the intersection of kernels of $n-d$ linear maps.
 A: Extend $e_1, \ldots, e_d$ to a basis $e_1, \ldots, e_n$ of $F_q^n$. For each $i$ between $1$ and $n$, define a linear form $e_i^*$ by its action on the basis: let $e_i^*(e_j)$ be $0$ when $i \neq j$ and $1$ when $i = j$. The linear forms $e_1^*, \ldots, e_n^*$ form the dual basis to the basis $e_1, \ldots, e_n$.
I claim that $\mathcal{C} = \bigcap_{i = d + 1}^n \ker e^*_i$. Note that if $1 \le j \le d$ and $d + 1 \le i \le n$, then
$$e_i^*(e_j) = 0 \implies e_j \in \ker e_i^*,$$
hence
$$\mathcal{C} = \operatorname{span}(e_1, \ldots, e_d) \subseteq \bigcap_{i=d+1}^n \ker e_i^*.$$
Conversely, suppose $x \in \bigcap_{i=d+1}^n e_i^*$. Since $x \in F_q^n$, we have $x = a_1 e_1 + \ldots + a_n e_n$ for some scalars $a_1, \ldots, a_n \in F_q$. We have, for $d + 1 \le i \le n$,
$$0 = e_i^*(x) = a_1 e_i^*(e_1) + \ldots + a_{i - 1} e_i^*(e_{i - 1}) + a_i e_i^*(e_i) + a_{i + 1} e_i^*(e_{i + 1}) + \ldots + a_n e_i^*(e_n) = a_i,$$
hence
$$x = a_1 e_1 + \ldots + a_d e_d + 0 + \ldots + 0 \in \mathcal{C}.$$
Thus, $\mathcal{C}$ can indeed be expressed as the intersection of the kernels of $n - d$ linear forms.
