# Connexions

You are here: Home » Content » Projection Theorem

### Recently Viewed

This feature requires Javascript to be enabled.

# Projection Theorem

Module by: Phil Schniter. E-mail the author

Summary: This module introduces projection theorem.

• The orthogonal projection of y onto S, where SS is a closed subspace of VV, is y ̂ = i i( x i ,y) x i y ̂ i i x i y x i s.t. x i x i is an ON basis for SS. Orthogonal projection yields the best approximation of yy in SS: y ̂ =argmin xS yx y ̂ argmin x S y x The approximation error e=y y ̂ e y y ̂ obeys to orthogonality principle: eS e S We illustrate this concept using V=R3 V 3 below but stress that the same geometrical interpretation applies to any Hilbert space.

A proof of the orthogonality principle is: eS i :e, x i =0 e S i e x i 0 y y ̂ , x i =0 y y ̂ x i 0 y, x i = y ̂ , x i y x i y ̂ x i y, x i = j j( x j ,y) x j , x i y x i j j x j y x j x i y, x i = j j x j ,y¯( x j , x i ) y x i j j x j y x j x i y, x i = j j(y, x j ) d i j y x i j j y x j d i j y, x i =y, x i y x i y x i

## Content actions

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks