Page 584 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Let be the vector space of functions with continuous first derivatives on , and let
be the
be the vector space of all real-valued functions defined on . Let

transformation that maps a function into its derivative—that is,

From the properties of differentiation, we have
Thus, D is a linear transformation.

EXAMPLE 12 A Linear Transformation from to

Calculus Required

Let be the vector space of continuous functions on , and let be the

vector space of functions with continuous first derivatives on . Let be the transformation that maps

into the integral . For example, if , then

From the properties of integration, we have

so J is a linear transformation.

EXAMPLE 13 A Transformation That Is Not Linear
Let be the transformation that maps an matrix into its determinant:

If , then this transformation does not satisfy either of the properties required of a linear transformation. For example, we
saw in Example 1 of Section 2.3 that

in general. Moreover, , so

in general. Thus T is not a linear transformation.

Properties of Linear Transformations

If is a linear transformation, then for any vectors and in V and any scalars and , we have