Linear and multilinear algebra; matrix theory Definition

Orthogonal complement

Builds on Orthogonal vectors Linear subspace

Definition

Let $V$ be an inner product space over $\R$ or $\C$ , and let $W$ be a subspace of $V$ . The orthogonal complement of $W$ is $W^\perp = \{v \in V : \langle v, w \rangle = 0 \text{ for all } w \in W\}.$

Remarks

$W^\perp$ is a subspace of $V$ , and $W \cap W^\perp = 0$ . In $\R^3$ with the dot product, the orthogonal complement of a plane through $0$ is its normal line. For a real $m \times n$ matrix $A$ , the null space of $A$ is the orthogonal complement of the row space of $A$ in $\R^n$ . In finite dimensions, $V = W \oplus W^\perp$ , see the orthogonal decomposition.

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Vector space definition
Inner product definition
Linear subspace definition
Orthogonal vectors definition
Orthogonal complement definition

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39 pages build on this one.

Tag	`079D68S3AW`
URI	`m2://079D68S3AW`
Kind	definition
Writing	W1 stated
MSC	15A63, 46C05
Defines	Orthogonal complement
Uses	Orthogonal vectors, Linear subspace

Orthogonal complement

Definition

Remarks

Reading path

Where this leads

Further reading

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