Page 529 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

4.
(a) Show that if A is orthogonal, then is orthogonal.

(b) What is the normal system for when A is orthogonal?

Verify that the reflection matrices in Tables 2 and 3 of Section 4.2 are orthogonal.
5.

Let a rectangular -coordinate system be obtained by rotating a rectangular -coordinate system counterclockwise through

6. the angle .

(a) Find the -coordinates of the point whose -coordinates are (−2, 6).
(b) Find the -coordinates of the point whose -coordinates are (5, 2).

Repeat Exercise 6 with .
7.

Let a rectangular -coordinate system be obtained by rotating a rectangular -coordinate system counterclockwise about

8. the z-axis (looking down the z-axis) through the angle .

(a) Find the -coordinates of the point whose -coordinates are (−1, 2, 5).

(b) Find the -coordinates of the point whose -coordinates are ( 1, 6, −3).

Repeat Exercise 8 for a rotation of counterclockwise about the y-axis (looking along the positive y-axis toward the
9. origin). counterclockwise about the x-axis (looking along the positive x-axis toward the

Repeat Exercise 8 for a rotation of
10. origin).

11.

(a) A rectangular -coordinate system is obtained by rotating an -coordinate system counterclockwise about the

y-axis through an angle (looking along the positive y-axis toward the origin). Find a matrix A such that

where and are the coordinates of the same point in the - and -systems

respectively.