Page 286 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Table 6 Figure 4.2.4 Equations Standard Matrix
Illustration
Operator

Rotation through an angle

Then, from basic trigonometry, (14)
and (15)
Using trigonometric identities on 15 yields

and substituting14 yields

(16)

The equations in 16 are linear, so T is a linear operator; moreover, it follows from these equations that the standard matrix for T
is

EXAMPLE 5 Rotation , then the image w of a vector
If each vector in is rotated through an angle of

is