Rongzhou at 04:38, 13 October 2026

2026-10-13T04:38:46Z

← Older revision		Revision as of 04:38, 13 October 2026
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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 ==

Rongzhou: 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..."

2026-10-12T11:10:16Z

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..."

New page

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

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

It can be equivalently stated that

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

<blockquote>
Intuition: recall an [[inner product]] is linear in the first slot and conjugately symmetric
</blockquote>

Hyperplane - Revision history

Rongzhou at 04:38, 13 October 2026