Definition:Orthogonal (Bilinear Form)

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This page is about Orthogonal (Bilinear Form). For other uses, see Orthogonal.

Definition

Let $\mathbb K$ be a field.

Let $V$ be a vector space over $\mathbb K$.

Let $b: V \times V \to \mathbb K$ be a reflexive bilinear form on $V$.

Let $v,w\in V$.


Then $v$ and $w$ are orthogonal (with respect to $b$) if and only if $\map b {v, w} = \map b {w, v} = 0$


This is denoted: $v \perp w$.


Orthogonal Subsets

Let $S, T \subset V$ be subsets.


Then $S$ and $T$ are orthogonal if and only if for all $s \in S$ and $t \in T$, $s$ and $t$ are orthogonal: $s \perp t$.


Orthogonal Complement

Let $S\subset V$ be a subset.


The orthogonal complement of $S$ (with respect to $b$) is the set of all $v \in V$ which are orthogonal to all $s \in S$.


This is denoted: $S^\perp$.

If $S = \set v$ is a singleton, we also write $v^\perp$.


Radical

The radical of $V$ is the orthogonal complement of $V$:

$\map {\operatorname {rad} } V = V^\perp$


Also see


Sources