Definition:Orthogonal (Bilinear Form)/Orthogonal Complement

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 $S\subset V$ 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 $v^\perp$.