# projection

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

$${P}^{2}=P.$$ | (1) |

###### Proposition 1

If $P\mathrm{:}V\mathrm{\to}V$ is a projection, then its image and the kernel are complementary subspaces, namely

$$V=\mathrm{ker}P\oplus imgP.$$ | (2) |

*Proof. *
Suppose that $P$ is a projection. Let $v\in V$ be given, and set

$$u=v-Pv.$$ |

The projection condition (1) then implies that $u\in \mathrm{ker}P$, and we can write $v$ as the sum of an image and kernel vectors:

$$v=u+Pv.$$ |

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 $P$.
Then, $Pv=v$ and $Pv=0$, and hence $v=0$. QED

Conversely, every direct sum^{} decomposition

$$V={V}_{1}\oplus {V}_{2}$$ |

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

$$Pv=\{\begin{array}{cc}v\hfill & v\in {V}_{1}\hfill \\ 0\hfill & v\in {V}_{2}\hfill \end{array}$$ |

Specializing somewhat, suppose that the ground field is $\mathbb{R}$ or
$\u2102$ and that $V$ is equipped with a positive-definite inner
product. In this setting we call an endomorphism^{}
$P:V\to V$ an *orthogonal projection* if it is self-dual

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

###### Proposition 2

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

*Proof. *
Let $u\in \mathrm{ker}P$ and $v\in imgP$ be given. Since $P$
is self-dual we have

$$0=\u27e8Pu,v\u27e9=\u27e8u,Pv\u27e9=\u27e8u,v\u27e9.$$ |

QED

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

$$\u27e8v-Pv,u\u27e9=0$$ |

for all $u\in imgP$.

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 |