Linear and multilinear algebra; matrix theory Definition

Projection

Builds on Endomorphism Direct sum of subspaces

Definition

Let $F$ be a field and let $V$ be a vector space over $F$ . A projection on $V$ is an endomorphism $P \colon V \to V$ with $P \circ P = P$ .

Remarks

If $P$ is a projection, then $V = \im P \oplus \ker P$ , and $P$ acts as the identity on $\im P$ . Conversely, for each direct sum $V = U \oplus W$ , the map $u + w \mapsto u$ is a projection with image $U$ and kernel $W$ . If $P$ is a projection, then $I - P$ is the projection with image $\ker P$ and kernel $\im P$ . In an inner product space, an orthogonal projection is a projection whose kernel is orthogonal to its image.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Vector space definition
Linear map definition
Linear subspace definition
Endomorphism definition
Sum of subspaces definition
Direct sum of subspaces definition
Projection definition

Where this leads

35 pages build on this one.

Tag	`0RH4E3CAV9`
URI	`m2://0RH4E3CAV9`
Kind	definition
Writing	W1 stated
MSC	15A04
Defines	Projection
Uses	Endomorphism, Direct sum of subspaces

Projection

Definition

Remarks

Reading path

Where this leads

Further reading

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