Page 592 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

(c) be the orthogonal projection of onto the -plane. Show that .

Let
25.

26. be a linear transformation, and let k be a scalar. Define the function by
(a) Let . Show that is a linear transformation.

(b) Find if is given by the formula .

27. and be linear transformations. Define the functions and
(a) Let by

Show that and are linear transformations. are given by the formulas
(b) Find
and if and
and .

28.
(a) Prove that if , , , and are any scalars, then the formula

defines a linear operator on . define a linear operator on ? Explain.
(b) Does the formula

Let be a basis for a vector space V, and let be a linear transformation. Show that if
29.
, then T is the zero transformation.
Let
30. be a basis for a vector space V, and let be a linear operator. Show that if ,

,…, , then T is the identity transformation on V.

31. (For Readers Who Have Studied Calculus) Let