A linear transformation P:VV of a vector spaceMathworldPlanetmath V is called a projection if it acts like the identityPlanetmathPlanetmathPlanetmath on its image. This condition can be more succinctly expressed by the equation

P2=P. (1)
Proposition 1

If P:VV is a projection, then its image and the kernel are complementary subspaces, namely

V=kerPimgP. (2)

Proof. Suppose that P is a projection. Let vV be given, and set


The projection condition (1) then implies that ukerP, and we can write v as the sum of an image and kernel vectors:


This decomposition is unique, because the intersectionMathworldPlanetmath of the image and the kernel is the trivial subspacePlanetmathPlanetmathPlanetmath. Indeed, suppose that vV is in both the image and the kernel of P. Then, Pv=v and Pv=0, and hence v=0. QED

Conversely, every direct sumMathworldPlanetmath decomposition


corresponds to a projection P:VV defined by


Specializing somewhat, suppose that the ground field is or and that V is equipped with a positive-definite inner product. In this setting we call an endomorphismPlanetmathPlanetmathPlanetmath P:VV an orthogonal projection if it is self-dual


in additionPlanetmathPlanetmath to satisfying the projection condition (1).

Proposition 2

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

Proof. Let ukerP and vimgP be given. Since P is self-dual we have



Thus we see that a orthogonal projection P projects a vV onto Pv in an orthogonalMathworldPlanetmathPlanetmathPlanetmath fashion, i.e.


for all uimgP.

Title projection
Canonical name Projection
Date of creation 2013-03-22 12:52:13
Last modified on 2013-03-22 12:52:13
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 8
Author rmilson (146)
Entry type Definition
Classification msc 15A21
Classification msc 15A57
Defines orthogonal projection