PlanetMath: projection

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projection

(Definition)

A linear transformation $P:V\rightarrow V$ of a vector space is called a projection if it acts like the identity on its image. This condition can be more succinctly expressed by the equation

$\displaystyle P^2 = P.$

(1)

Proposition 1 If $P:V\rightarrow V$ is a projection, then its image and the kernel are complementary subspaces, namely

$\displaystyle V = \ker P \oplus \mathop{\mathrm{img}}\nolimits P.$

(2)

Proof. Suppose that

is a projection. Let $v\in V$ be given, and set

$\displaystyle u=v-Pv.$

The projection condition (1) then implies that $u\in \ker P$ , and we can write

as the sum of an image and kernel vectors:

$\displaystyle v = u + Pv.$

This decomposition is unique, because the intersection of the image and the kernel is the trivial subspace. Indeed, suppose that $v\in V$ is in both the image and the kernel of

. Then,

and

, and hence

. QED

Conversely, every direct sum decomposition

$\displaystyle V = V_1 \oplus V_2$

corresponds to a projection $P:V\rightarrow V$ defined by

$\displaystyle Pv =\begin{cases} v & v\in V_1 \ 0 & v\in V_2 \end{cases}$

Specializing somewhat, suppose that the ground field is $\mathbb{R}$ or $\mathbb{C}$ and that is equipped with a positive-definite inner product. In this setting we call an endomorphism $P:V\rightarrow V$ an orthogonal projection if it is self-dual

$\displaystyle P^{\displaystyle \star}= P,$

in addition to satisfying the projection condition (1).

Proposition 2 The kernel and image of an orthogonal projection are orthogonal subspaces.

Proof. Let $u\in\ker P$ and $v\in \mathop{\mathrm{img}}\nolimits P$ be given. Since

is self-dual we have

$\displaystyle 0 = \langle Pu,v\rangle = \langle u,Pv\rangle = \langle u,v\rangle.$

QED

Thus we see that a orthogonal projection projects a $v \in V$ onto in an orthogonal fashion, i.e.

$\displaystyle \langle v-Pv,u\rangle = 0$

for all $u\in \mathop{\mathrm{img}}\nolimits P$ .

"projection" is owned by rmilson.

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Also defines:

orthogonal projection

Attachments:: projection of point (Definition) by pahio

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Cross-references: onto, projects, orthogonal, addition, self-dual, endomorphism, inner product, ground field, direct sum, QED, subspace, intersection, decomposition, vectors, sum, implies, complementary subspaces, kernel, equation, image, identity, vector space, linear transformation
There are 52 references to this entry.

This is version 5 of projection, born on 2002-07-26, modified 2004-02-25.
Object id is 3208, canonical name is Projection.
Accessed 17249 times total.

Classification:

AMS MSC:	15A21 (Linear and multilinear algebra; matrix theory :: Canonical forms, reductions, classification)
	15A57 (Linear and multilinear algebra; matrix theory :: Other types of matrices )

Pending Errata and Addenda

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