Page 598 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Dimension Theorem for Linear Transformations

Recall from the Dimension Theorem for Matrices (Theorem 5.6.3) that if A is a matrix with n columns, then
The following theorem, whose proof is deferred to the end of the section, extends this result to general linear transformations.

THEOREM 8.2.3

Dimension Theorem for Linear Transformations
If is a linear transformation from an n-dimensional vector space V to a vector space W, then

(1)

In words, this theorem states that for linear transformations the rank plus the nullity is equal to the dimension of the domain.
This theorem is also known as the Rank Theorem.

Remark If A is an matrix and is multiplication by A, then the domain of has dimension n, so Theorem

8.2.3 agrees with Theorem 5.6.3 in this case.

EXAMPLE 9 Using the Dimension Theorem
Let be the linear operator that rotates each vector in the -plane through an angle . We showed in Example 5
that and . Thus

which is consistent with the fact that the domain of T is two-dimensional.

Additional Proof

Proof of Theorem 8.2.3 We must show that

We shall give the proof for the case where . The cases where and

are left as exercises. Assume , and let ,…, be a basis for the kernel.

Since is linearly independent, Theorem 5.4.6b states that there are vectors, , …,

, such that the extended set is a basis for V. To complete the proof, we shall

show that the vectors in the set form a basis for the range of T. It will then

follow that

First we show that S spans the range of T. If is any vector in the range of T, then for some vector in V. Since