Problems 3) In this problem, you will show that every 2 x 2 orthogonal matrix is either a rotation or a reflection. (a) Let Q = [a b be an orthogonal 2 x 2 matrix. Show that the following equations hold: a2 + 62 = 1 b2 + d2 = 1 ac + bd = 1 a2 + c2 = 1 c2 + d2 = 1 ab + cd = 1 Hint: Consider QT Q and QQT. (b) Whenever we have the equation of the form x2 + y2 = 1 we can find an angle 0 such that x = cos 0 and y = +sin 0 (i.e. either y = sin 0 ory = -sin 0 ). Using this fact and part (a), show that we can find 0 such that: a = cos 0 c = +sin 0 b = +sin 0 d = +cos 0 (c) Suppose we pick d = -cos 0 . What are the two possible matrices we get for Q? (d) Given a 2-vector x, let's try to visualize the effects of transforming x by Q. That is, let's visualize what changes between x and Qx. Compute the numerical value for one of the matrices Q you obtained in part (c) for 0 = 1. Give a one sentence interpretation of what is happening in the plot, and include a printout or drawing of the plot. (e) Suppose we pick d = cos 0 . What are the two possible matrices we get for Q? (f) Repeat part (d) using one of the two matrices you obtained in (e) and 0 = n/2. Give a one sentence interpretation of what is happening in the plot, and include a printout or drawing of the plot