Hyperplane

Let $V$ be a vector space over $\mathbb{F}$.

$$\begin{gathered}P=\left\{ \vec p + \sum_{i=1}^{k} \lambda_{i}\vec v_{i}\, |\, \vec p,\vec v_i \in V,k < \dim(V), \lambda_{i} \in \mathbb{F} \right\}, \\\end{gathered}$$

and $\{\vec v_i\}$ is linearly independent,

Then $P$ is a hyperplane of dimension $k$ in $V$

Properties

Inner product space

If $V$ is an inner product space over $\mathbb{F}$ and $P$ is a hyperplane of dimension $k$.

$\exists \vec w \in V, b \in \mathbb{F} \Rightarrow \forall \vec u \in P, \langle \vec w, \vec u \rangle =b$

Then $\vec w$ is called the normal vector of hyperplane $P$.

If it is known that $P=\left\{ \vec p + \sum_{i=1}^{k} \lambda_{i}\vec v_{i}\, |\, \vec p,\vec v_i \in V,k < \dim(V), \lambda_{i} \in \mathbb{F} \right\}$,

Then $b=\langle \vec w, \vec p \rangle$

It can be equivalently stated that

$$ \langle \vec w, \vec u - \vec p \rangle =0 $$

Intuition: recall an inner product is linear in the first slot and conjugately symmetric