Page 307 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

From a more geometric viewpoint, if w is the image of x under , then maps w back into x, since

(Figure 4.3.3).

Figure 4.3.3

Before turning to an example, it will be helpful to touch on a notational matter. When a one-to-one linear operator on is written

as (rather than ), then the inverse of the operator T is denoted by (rather than ). Since the

standard matrix for is the inverse of the standard matrix for T, we have

(1)

EXAMPLE 4 Standard Matrix for

Let be the operator that rotates each vector in through the angle , so from Table 6 of Section 4.2,

It is evident geometrically that to undo the effect of T, one must rotate each vector in through the angle (2)
what the operator does, since the standard matrix for is . But this is exactly

(verify), which is identical to 2 except that is replaced by .

EXAMPLE 5 Finding defined by the equations
Show that the linear operator

is one-to-one, and find .

Solution

The matrix form of these equations is