On-Line Computer Graphics Notes

Orthogonality Inner Product Spaces

Overview

In the usual linear space of vectors in three-dimensional space, we utilize the dot product to define when two vectors are perpendicular, or orthogonal. We can do the same thing in a general inner product space. These notes review this concept of orthogonality in inner product spaces.

Definition of Orthogonality

Given two elements f and g in an inner product space, they are said to be orthogonal if

We frequently utilize the synonym perpendicular for orthogonal.

Definition of Orthonormal

Given a set of elements , , ..., in an inner product space, they are said to be orthonormal if

Any finite number of orthonormal elements are linear independent, since if

then dotting both sides by any , we have

which implies that for each i.

The Gram-Schmidt Orthogonalization Process

Given a set of linearly independent elements from a linear space , we can produce a second set of elements that are orthonormal and whose span is the same as the original set.

This method proceeds as follows:

• Define .
• Define and .
• In general, having defined , , ..., , we define by first defining

  

  and then define

  

It should be clear to the reader that , because the process was constructed in this way. Also if we look at we see that

which implies that and at least form the start of an orthonormal set.

Inductively, if we assume that , , ..., and form an orthonormal set, then for j<k

and so the set , , ..., and forms an orthonormal set.

This document maintained by Ken Joy Comments to the Author All contents copyright (c) 1996, 1997 Computer Science Department, University of California, Davis All rights reserved.