Definition:Orthogonal (Linear Algebra)/Orthogonal Complement

Definition
Let $\mathbb K$ be a field.

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

Let $\struct {V, \innerprod \cdot \cdot}$ be an inner product space. Let $S\subseteq V$ be a subset.

The orthogonal complement of $S$ (with respect to $\innerprod \cdot \cdot$) 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 $v^\perp$.