Page 292 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Figure 4.2.7
Thus the standard matrices for these linear operators are

These matrices should satisfy 21. With the help of some basic trigonometric identities, we can show that this is so as follows:

Remark In general, the order in which linear transformations are composed matters. This is to be expected, since the
composition of two linear transformations corresponds to the multiplication of their standard matrices, and we know that the
order in which matrices are multiplied makes a difference.

EXAMPLE 7 Composition Is Not Commutative
Let be the reflection operator about the line , and let be the orthogonal projection on the
y-axis. Figure 4.2.8 illustrates graphically that and have different effects on a vector x. This same conclusion
can be reached by showing that the standard matrices for and do not commute:

so .