Page 604 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

27. (For Readers Who Have Studied Calculus) Let be the differentiation transformation , where

and . Describe the kernels of and

Fill in the blanks.
28.

(a) If is multiplication by A, then the nullspace of A corresponds to the

_________ of , and the column space of A corresponds to the _________ of .

(b) If is the orthogonal projection on the plane , then the kernel of

T is the line through the origin that is parallel to the vector _________ .

(c) If V is a finite-dimensional vector space and is a linear transformation, then

the dimension of the range of T plus the dimension of the kernel of T is _________ .

(d) If is multiplication by A, and if , then the general solution of
has _________ (howman y?) parameters.

29. is a linear operator, and if the kernel of T is a line through the origin, then
(a) If

what kind of geometric object is the range of T? Explain your reasoning.

(b) If is a linear operator, and if the range of T is a plane through the origin, then

what kind of geometric object is the kernel of T? Explain your reasoning.

30. (For Readers Who Have Studied Calculus) Let V be the vector space of real-valued functions

with continuous derivatives of all orders on the interval , and let

be the vector space of real-valued functions defined on .

(a) Find a linear transformation whose kernel is .

(b) Find a linear transformation whose kernel is .

If A is an matrix, and if the linear system is consistent for every vector in ,
?
31. what can you say about the range of

Copyright © 2005 John Wiley & Sons, Inc. All rights reserved.