Definition:Orthogonal (Linear Algebra)/Orthogonal Complement

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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 $v^\perp$.

Also known as

The orthogonal complement of a subset $A$ of an inner product space is also known as its orthocomplement.

The operation of assigning the orthogonal complement $A^\perp$ to $A$ is referred to as orthocomplementation.