Page 283 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Consider the operator that maps each vector into its symmetric image about the y-axis (Figure 4.2.2).

If we let Figure 4.2.2
, then the equations relating the components of x and w are

or, in matrix form, (10)
(11)

Since the equations in 10 are linear, T is a linear operator, and from 11 the standard matrix for T is

In general, operators on and that map each vector into its symmetric image about some line or plane are called reflection
operators. Such operators are linear. Tables 2 and 3 list some of the common reflection operators.

Table 2

Operator Illustration Equations Standard Matrix

Reflection about the y-axis

Reflection about the x-axis

Reflection about the line

Table 3