Page 662 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

22. (For Readers Who Have Studied Calculus)

Let and be continuous functions, and let V be the subspace of consisting of all twice

differentiable functions. Define by

(a) Show that L is a linear transformation.

(b) Consider the special case where and . Show that the function is in the

nullspace of L for all real values of and .

23. (For Readers Who Have Studied Calculus)

Let be the differentiation operator . Show that the matrix for D with respect to the basis

is

24. (For Readers Who Have Studied Calculus)
It can be shown that for any real number c, the vectors

form a basis for . Find the matrix for the differentiation operator of Exercise 23 with respect to this basis.

25. (For Readers Who Have Studied Calculus)
Let be the integration transformation defined by

where . Find the matrix for T with respect to the standard bases for and .

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