Page 657 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Chapter 8 a

Supplementary Exercises

Let A be an matrix, B a nonzero matrix, and x a vector in expressed in matrix notation. Is
1. linear operator on ? Justify your answer.

Let
2.

(a) Show that

(b) Guess the form of the matrix for any positive integer n.

(c) By considering the geometric effect of , where T is multiplication by A, obtain the result in (b)
geometrically.

Let be a fixed vector in an inner product space V, and let be defined by . Show that T is a
3. linear operator on V.

Let , ,…, be fixed vectors in , and let be the function defined by

4. , where is the Euclidean inner product on .

(a) Show that T is a linear transformation.

(b) Show that the matrix with row vectors , , … , is the standard matrix for T.

Let be the standard basis for , and let be the linear transformation for which
5.

(a) Find bases for the range and kernel of T.