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(c) In the -plane, a rotation about the origin followed by a reflection about a coordinate
axis is one-to-one.

Does the formula define a one-to-one linear transformation from
25. to ? Explain your reasoning.

Let E be a fixed elementary matrix. Does the formula define a one-to-one
26. linear operator on ? Explain your reasoning.

Let be a fixed vector in . Does the formula define a one-to-one linear
27. operator on ? Explain your reasoning.

28. (For Readers Who Have Studied Calculus) The Fundamental Theorem of Calculus implies

that integration and differentiation reverse the actions of each other in some sense. Define a

transformation by , and define by

.

(a) Show that D and J are linear transformations.

(b) Explain why J is not the inverse transformation of D.

(c) Can we restrict the domains and/or codomains of D and J such that they are inverse
linear transformations of each other?

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