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.

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.

Also see

• Results about orthogonal complements can be found here.

Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): $\text{I}$ Hilbert Spaces: $\S 2.$ Orthogonality
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): orthogonal complement
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): orthogonal complement
• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Definitions