Definition:Orthogonal (Bilinear Form)

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 $b \left({v, w}\right) = b \left({w, v}\right) = 0$

This is denoted: $v \perp w$.

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: $s \perp t$.

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 $S = \left\{ {v}\right\}$ is a singleton, we also write $v^\perp$.

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

$\operatorname{rad}(V) = V^\perp$