Problem 1. Let O2(R) = {Q E M2(R) : QTQ = /} be the group of 2 x 2 orthogonal matrices, and fix Q E O2(R). 1.1. Show that det (Q) = 1 or det(Q) = -1. 1.2. Show that the transformation To : R2 -> R2 defined by To(v) = Qu is a symmetry of the unit circle S = {v E R2 : | |v|| = 1}. 1.3. Show that (Qu) . (Qw) = v . w for all v, w E R2. Here . is the dot product, defined by v . w = vw. 1.4. It is general fact from linear algebra that the angle 0