Page 617 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

20. be the integration transformation .
(For Readers Who Have Studied Calculus) Let
Determine whether J is one-to-one. Justify your conclusion.

21. (For Readers Who Have Studied Calculus) Let V be the vector space and let be defined by

. Verify that T is a linear transformation. Determine whether T is one-to-one. Justify

your conclusion.

In Exercises 22 and 23, determine whether .

22. is the orthogonal projection on the x-axis, and is the orthogonal projection on the
(a)

y-axis.

(b) is the rotation about the origin through an angle , and is the rotation about the
origin through an angle .

(c) is the rotation about the x-axis through an angle , and is the rotation about the
z-axis through an angle .

23. is the reflection about the x-axis, and is the reflection about the y-axis.
(a)

(b) is the orthogonal projection on the x-axis, and is the counterclockwise rotation
through an angle .

(c) is a dilation by a factor k, and is the counterclockwise rotation about the z-axis
through an angle .

Indicate whether each statement is always true or sometimes false. Justify your answer by
24. giving a logical argument or a counterexample.

(a) If is the orthogonal projection onto the x-axis, then

maps each point on the x-axis onto a line that is perpendicular to the x-axis.

(b) If and are linear transformations, and if is not
.
one-to-one, then neither is