Page 594 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

8.2 In this section we shall develop some basic properties of linear
transformations that generalize properties of matrix transformations obtained
KERNEL AND RANGE earlier in the text.

Kernel and Range

Recall that if A is an matrix, then the nullspace of A consists of all vectors in such that , and by Theorem 5.5.1

the column space of A consists of all vectors in for which there is at least one vector in such that . From the

viewpoint of matrix transformations, the nullspace of A consists of all vectors in that multiplication by A maps into 0, and

the column space of A consists of all vectors in that are images of at least one vector in under multiplication by A. The

following definition extends these ideas to general linear transformations.

DEFINITION

If is a linear transformation, then the set of vectors in V that T maps into 0 is called the kernel of T; it is denoted
by ker(T). The set of all vectors in W that are images under T of at least one vector in V is called the range of T; it is denoted
by .

EXAMPLE 1 Kernel and Range of a Matrix Transformation

If is multiplication by the matrix A, then from the discussion preceding the definition above, the kernel of

is the nullspace of A, and the range of is the column space of A.

EXAMPLE 2 Kernel and Range of the Zero Transformation

Let be the zero transformation (Example 2 of Section 8.1). Since T maps every vector in V into 0, it follows that

. Moreover, since 0 is the only image under T of vectors in V, we have .

EXAMPLE 3 Kernel and Range of the Identity Operator

Let be the identity operator (Example 3 of Section 8.1). Since for all vectors in V, every vector in V is the

image of some vector (namely, itself); thus . Since the only vector that I maps into 0 is 0, it follows that .