Page 675 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

If is a compression or expansion in the x-direction with factor k, then
so the standard matrix for T is
Similarly, the standard matrix for a compression or expansion in the y-direction is

EXAMPLE 1 Operating with Diagonal Matrices
Suppose that the -plane first is compressed or expanded by a factor of in the x-direction and then is compressed or expanded
by a factor of in the y-direction. Find a single matrix operator that performs both operations.

Solution

The standard matrices for the two operations are

Thus the standard matrix for the composition of the x-operation followed by the y-operation is

(2)

This shows that multiplication by a diagonal matrix compresses or expands the plane in the x-direction and also in the

y-direction. In the special case where and are the same, say , note that 2 simplifies to

which is a contraction or a dilation (Table 8 of Section 4.2).

Shears

A shear in the x-direction with factor k is a transformation that moves each point parallel to the x-axis by an amount to

the new position . Under such a transformation, points on the x-axis are unmoved since . However, as we

progress away from the x-axis, the magnitude of y increases, so points farther from the x-axis move a greater distance than those

closer (Figure 9.2.3).