Page 284 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Operator Illustration Equations Standard Matrix

Reflection about the -plane

Reflection about the -plane
Reflection about the -plane

Projection Operators

Consider the operator that maps each vector into its orthogonal projection on the x-axis (Figure 4.2.3). The

equations relating the components of x and are

(12)

or, in matrix form,

(13)

Figure 4.2.3

The equations in 12 are linear, so T is a linear operator, and from 13 the standard matrix for T is

In general, a projection operator (more precisely, an orthogonal projection operator) on or is any operator that maps
each vector into its orthogonal projection on a line or plane through the origin. It can be shown that such operators are linear.
Some of the basic projection operators on and are listed in Tables 4 and 5.