Page 597 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

DEFINITION

If is a linear transformation, then the dimension of the range of T is called the rank of T and is denoted by

; the dimension of the kernel is called the nullity of T and is denoted by .

If A is an matrix and is multiplication by A, then we know from Example 1 that the kernel of is the

nullspace of A and the range of is the column space of A. Thus we have the following relationship between the rank and

nullity of a matrix and the rank and nullity of the corresponding matrix transformation.

THEOREM 8.2.2

If A is an matrix and is multiplication by A, then
(a)

(b)

EXAMPLE 7 Finding Rank and Nullity
Let be multiplication by

Find the rank and nullity of . and . Thus, from Theorem 8.2.2, we have

Solution

In Example 1 of Section 5.6, we showed that rank
and .

EXAMPLE 8 Finding Rank and Nullity

Let be the orthogonal projection on the -plane. From Example 4, the kernel of T is the z-axis, which is
one-dimensional, and the range of T is the -plane, which is two-dimensional. Thus