Page 583 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

EXAMPLE 8 A Linear Transformation from to by
Let be a polynomial in , and define the function

The function T is a linear transformation, since for any scalar k and any polynomials and in we have

and
(Compare this to Exercise 4 of Section 4.4.)

EXAMPLE 9 A Linear Operator on

Let be a polynomial in , and let a and b be any scalars. We leave it as an exercise to
show that the function T defined by

is a linear operator. For example, if , then would be the linear operator given by the formula

EXAMPLE 10 A Linear Transformation Using an Inner Product be the transformation that maps a vector
Let V be an inner product space, and let be any fixed vector in V. Let
v into its inner product with —that is,

From the properties of an inner product,

and

so T is a linear transformation.

EXAMPLE 11 A Linear Transformation from to

Calculus Required