Prove every subspace of finite dim V is the intersection of hyperplanes

Question

A hyperplane in V is defined as the kernel of a linear functional. Show that every subspace of V is the intersection of hyperplanes.

Please can someone offer insight into how I prove this?

Thanks a lot!

Answer 1

Hint: Let $W \subset V$ be a subspace, let $\{v_1, \ldots, v_k\}$ be a basis for $W$, and extend it to a basis $\{v_1, \ldots, v_k, \ldots, v_n\}$ of $V$.

For each $i \in \{1, 2, \ldots, n\}$, let $f_i: V \to F$ be the linear functional defined by $v_i \mapsto 1$ and $v_j \mapsto 0$ for $j \neq i$. Use these functions to express $W$ as an intersection of hyperplanes.

Answer 2

One can makeup a little bit the notation considering an orthonormal basis $\{v_{1},\dots, v_{k}\}$ of $W$; then extend this to an orthonormal basis $\{v_{1}, \dots, v_{k}, \dots v_{n}\}$ of $V$; and consider the $n-k$ hyperplanes $$ H_{i} = \{x \in V : \langle x, v_{i}\rangle = 0\}, \quad \text{for} \quad k+1\leq i \leq n. $$ Let us write $x = \langle x, v_{1} \rangle + \cdots +\langle x, v_{k}\rangle + \cdots + \langle x, v_{n}\rangle \in V$. Then $$ \begin{align*} x \in W &\iff \langle x, v_{i}\rangle = 0 \quad \text{for} \quad k+1 \leq i \leq n \\ &\iff x \in H_{i} \quad \text{for} \quad k + 1\leq i \leq n\\ &\iff x \in \bigcap_{i= k+1}^{n}H_{i}. \end{align*} $$