Page 673 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

9.2 In Section 4.2 we studied some of the geometric properties of linear operators
on and . In this section we shall study linear operators on in a little
GEOMETRY OF LINEAR more depth. Some of the ideas that will be developed here have important
OPERATORS ON R2 applications to the field of computer graphics.

Vectors or Points (1)

If is the matrix operator whose standard matrix is

then

There are two equally good geometric interpretations of this formula. We may view the entries in the matrices

either as components of vectors or as coordinates of points. With the first interpretation, T maps arrows to arrows, and with the
second, points to points (Figure 9.2.1). The choice is a matter of taste.

Figure 9.2.1

In this section we shall view linear operators on as mapping points to points. One useful device for visualizing the behavior of a
linear operator is to observe its effect on the points of simple figures in the plane. For example, Table 1 shows the effect of some
basic linear operators on a unit square that has been partially colored.

Table 1