Page 280 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

transformation from to . If we denote this transformation by T, then and

EXAMPLE 1 A Transformation from to
The equations

define a transformation . With this transformation, the image of the point is

Thus, for example,

Linear Transformations from to

In the special case where the equations in 1 are linear, the transformation defined by those equations is called a
is defined by equations of the
linear transformation (or a linear operator if ). Thus a linear transformation

form

(2)

or, in matrix notation,

(3)

or more briefly by

(4)

The matrix is called the standard matrix for the linear transformation T, and T is called multiplication by A.

EXAMPLE 2 A Linear Transformation from to

The linear transformation defined by the equations

(5)

can be expressed in matrix form as