Page 291 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

to an observer below the South Pole it will appear clockwise. Thus, when a rotation in is described as clockwise or
counterclockwise, a direction of view along the axis of rotation must also be stated.

There are some other facts about the Earth's rotation that are useful for understanding general rotations in . For example, as

the Earth rotates about its axis, the North and South Poles remain fixed, as do all other points that lie on the axis of rotation.
Thus, the axis of rotation can be thought of as the line of fixed points in the Earth's rotation. Moreover, all points on the
Earth that are not on the axis of rotation move in circular paths that are centered on the axis and lie in planes that are
perpendicular to the axis. For example, the points in the Equatorial Plane move within the Equatorial Plane in circles about
the Earth's center.

Compositions of Linear Transformations

If and are linear transformations, then for each x in one can first compute , which is a

vector in , and then one can compute , which is a vector in . Thus, the application of followed by

produces a transformation from to . This transformation is called the composition of with and is denoted by

(read “ circle ”). Thus

(18)

The composition is linear since

(19)

so is multiplication by , which is a linear transformation. Formula 19 also tells us that the standard matrix for
is . This is expressed by the formula

(20)

Remark Formula 20 captures an important idea: Multiplying matrices is equivalent to composing the corresponding linear
transformations in the right-to-left order of the factors.

There is an alternative form of Formula 20: If and are linear transformations, then because the

standard matrix for the composition is the product of the standard matrices of and T, we have

(21)

EXAMPLE 6 Composition of Two Rotations

Let and be the linear operators that rotate vectors through the angles and , respectively. Thus
the operation

first rotates x through the angle , then rotates through the angle . It follows that the net effect of is to rotate

each vector in through the angle (Figure 4.2.7).