Page 308 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

so the standard matrix for T is
This matrix is invertible (so T is one-to-one) and the standard matrix for is

Thus

from which we conclude that

Linearity Properties

In the preceding section we defined a transformation to be linear if the equations relating x and are linear

equations. The following theorem provides an alternative characterization of linearity. This theorem is fundamental and will be the

basis for extending the concept of a linear transformation to more general settings later in this text.

THEOREM 4.3.2

Properties of Linear Transformations

A transformation is linear if and only if the following relationships hold for all vectors u and v in and for
every scalar c.

(a)

(b)

Proof Assume first that T is a linear transformation, and let A be the standard matrix for T. It follows from the basic arithmetic
properties of matrices that

and

Conversely, assume that properties (a) and (b) hold for the transformation T. We can prove that T is
linear by finding a matrix A with the property that