Prove that every hyperplane is a null space of a linear functional

Question

How can we prove that every hyperplane (a subspace of dimension $n-1$) is a null space of a linear functional? I don't know how to prove this. I tried a lot, but something is missing.

Answer 1

Assuming by 'hyperspace' you mean a $n-1$ dimensional subspace of an $n$-dimensional vector space:

Let $f$ be a non-zero linear functional $f:V\rightarrow \mathbb{F}$ where $V$ is a finite-dimensional vector space over the field $\mathbb{F}$; the rank of $f$ is 1 since it is a non-zero subspace of $\mathbb{F}$, and thus is $\mathbb{F}$. Thus, the nullspace of $f$, denoted $N_f$, fulfills $\dim N_f= \dim V-1$. Letting $\dim V=n$, we see that the nullspace of $f$ has dimension $n-1$; we say that a subspace of dimension $n-1$ is a hyperspace of $V$. Furthermore, let $H$ be a hyperspace of $V$ with basis $\{\alpha_1,\ldots, \alpha_{n-1}\}$, and then let $\alpha_n\notin H$, letting $\mathcal{B}=\{\alpha_1,\ldots, \alpha_{n-1},\alpha_n\}$ be an ordered basis of $V$. Then if $\alpha\in H$, we know that $\alpha=\sum_{i=1}^n{c_i\alpha_i}$ for $c_i\in \mathbb{F}$, and $c_n=0_\mathbb{F}$. We then let $f$ be any linear functional such that $f(\alpha_i)=0_\mathbb{F}$ for $1\leq i \leq n-1$, and then $f(\alpha_n)=c$ for any non-zero $c\in \mathbb{F}$. Thus, $H\subset V$ is the nullspace of $f$, and a subspace $H$ is a hyperspace iff it is the nullspace of some non-zero linear functional on $V$.

Answer 2

Hint: prove more generally that the map

$$W \mapsto \text{Ann }W = \{\lambda \in V^* : \lambda(W)=0\}$$

sets up a one-to-one, inclusion-reversing correspondence between the subspaces of $V$ and the subspaces of $V^*$. Try to write down its inverse explicitly. It sends a subspace of dimension $k$ to a subspace of dimension $n-k$.

In particular, hyperplanes of $V$ are mapped to lines of $V^*$. What does that tell you?