Page 305 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

EXAMPLE 1 One-to-One Linear Transformations

In the terminology of the preceding definition, the rotation operator of Figure 4.3.1 is one-to-one, but the orthogonal projection
operator of Figure 4.3.2 is not.

Let A be an matrix, and let be multiplication by A. We shall now investigate relationships between the
invertibility of A and properties of .

Recall from Theorem 2.3.6 (with w in place of b) that the following are equivalent:

A is invertible.

is consistent for every matrix w.

has exactly one solution for every matrix w.
However, the last of these statements is actually stronger than necessary. One can show that the following are equivalent (Exercise
24):

A is invertible.

is consistent for every matrix w.

has exactly one solution when the system is consistent.

Translating these into the corresponding statements about the linear operator , we deduce that the following are equivalent:
A is invertible.

For every vector w in , there is some vector x in such that . Stated another way, the range of is all of .

For every vector w in the range of , there is exactly one vector x in such that . Stated another way, is
one-to-one.
In summary, we have established the following theorem about linear operators on .

THEOREM 4.3.1

Equivalent Statements is multiplication by A, then the following statements are equivalent.
If A is an matrix and