Hyperplane: Difference between revisions
Created page with "Let $V$ be a vector space over $\mathbb{F}$. {{#content:Q2146}} == 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}\, |\, \v..." |
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Let $V$ be a vector space over $\mathbb{F}$. | Let $V$ be a vector space over $\mathbb{F}$. | ||
{{#content:Q2146}} | {{#content:Q2146}}<ref>{{#cite:Q2149}}</ref> | ||
== Properties == | == Properties == | ||
Latest revision as of 04:38, 13 October 2026
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 $V$[1]
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