Hyperplanes in $F^n$ I was asked to prove that a subset $H  = \{(x_1,\ldots,x_n) ∈ F^n\,|\,a_1x_1 + a_2x_2 + \ldots + a_nx_n = 0\}$ of $F^n$ is a hyperplane if and only if $a_i \ne 0$ for some $i$.
I did this by saying: IFF $\dim(H) = n - 1 \; \implies \; \text{rank}(a_1,\ldots,a_n) = 1 \; \implies \; (a_1, ..., a_n)$ is not the $0$ vector.
Now I'm being asked to show that all hyperplanes in $F^n$ have that form? I was given a hint that one can find $a_1,\ldots,a_n$ as a solution of a homogeneous system of $n − 1$ equations in $n$ variables, but I'm not really sure how to use this/set this proof up.
 A: Let $H \subseteq \Bbb{F}^n$ be a hyperplane, i.e. a subspace of dimension $n-1$. Pick a basis $\{v_1, \ldots, v_{n-1}\}$ for $H$ and consider the homogeneous $(n-1)\times n$ linear system
$$\begin{bmatrix} v_1(1) & v_1(2) & \cdots & v_1(n) \\ v_2(1) & v_2(2) & \cdots & v_2(n) \\ \vdots \\ v_{n-1}(1) & v_{n-1}(2) & \cdots & v_{n-1}(n)\end{bmatrix}\begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n\end{bmatrix} = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_{n-1}\end{bmatrix}\begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n\end{bmatrix} = 0.$$
The matrix has maximal rank $n-1$ so the space of solutions has dimension $n-(n-1)=1$.
Therefore there is a nonzero solution $(a_1, \ldots, a_n) \in \Bbb{F}^n$. Define a hyperplane $$G = \{(x_1, \ldots, x_n) \in \Bbb{F}^n : a_1x_1 + \cdots +a_nx_n = 0\}$$
and notice that for every $1 \le i \le n-1$ we have
$$v_i(1)a_1 + \cdots + v_i(n)a_n = 0$$
so $v_i \in G$. Hence $H=\operatorname{span}\{v_1, \ldots, v_{n-1}\}\subseteq G$. Since both $H$ and $G$ have dimension $n-1$, we conclude $H=G$.
