Page 595 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

EXAMPLE 4 Kernel and Range of an Orthogonal Projection

Let be the orthogonal projection on the -plane. The kernel of T is the set of points that T maps into

; these are the points on the z-axis (Figure 8.2.1a). Since T maps every point in into the -plane, the range of T must be

some subset of this plane. But every point in the -plane is the image under T of some point; in fact, it is the image

of all points on the vertical line that passes through (Figure 8.2.1b). Thus is the entire -plane.

Figure 8.2.1 is the z-axis. (b) is the entire -plane.
(a)

EXAMPLE 5 Kernel and Range of a Rotation

Let be the linear operator that rotates each vector in the -plane through the angle . (Figure 8.2.2). Since every

vector in the -plane can be obtained by rotating some vector through the angle . (why?), we have . Moreover, the

only vector that rotates into 0 is 0, so .

Figure 8.2.2
CEaXlcAuMluPsLREeq6uireKdernel of a Differentiation Transformation