Page 677 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

EXAMPLE 2 Finding Matrix Transformations

(a) Find a matrix transformation from to that first shears by a factor of 2 in the x-direction and then reflects about .

(b) Find a matrix transformation from to that first reflects about and then shears by a factor of 2 in the x-direction.

Solution (a)

The standard matrix for the shear is

and for the reflection is

Thus the standard matrix for the shear followed by the reflection is

Solution (b)

The reflection followed by the shear is represented by

In the last example, note that , so the effect of shearing and then reflecting is different from the effect of reflecting and

then shearing. This is illustrated geometrically in Figure 9.2.4, where we show the effects of the transformations on a unit square.