\begin{definition}[Definition:Orthogonal (Linear Algebra)/Orthogonal Complement]
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.
We define the '''orthogonal complement of $S$ (with respect to $\innerprod \cdot \cdot$)''', written $S^\perp$ as the set of all $v \in V$ which are orthogonal to all $s \in S$.
That is: 
:$S^\perp = \set {v \in V : \innerprod v s = 0 \text { for all } s \in S}$
If $S = \set v$ is a singleton, we may write $S^\perp$ as $v^\perp$.
\end{definition}