Consider the linear transformation T : R3 R3 whose standard matrix is A = 16 1 3 + 6 6 1

QUESTION:

Consider the linear transformation \(T:\mathbb{R}^3\rightarrow \mathbb{R}^3\) whose standard matrix is

a. Find a nonzero vector \(\mathrm{v_1}\) satisfying \(\mathrm{Av_1=v_1}\). (Hint: Proceed as in Exercise 1.4.5)

b. Find an orthonormal basis \(\left \{ \mathrm{v_2,v_3} \right \}\) for the plane orthogonal to \(\mathrm{v_1}\).

c. Let B= \(\left \{ \mathrm{v_1,v_2,v_3} \right \}\). Apply the change-of-basis formula to find \(\left [ T \right ]_B\) .

d. Use your answer to part c to explain why T is a rotation. (Also see Example 6 in Section 1 of Chapter 7.)