1. Cyclic property of the trace. (30 points) Recall from HW 3 that the trace of a matrix M is the sum of the diagonal entries, which in index notation is M'. The symbol for the trace of M is Tr( M). (a) Show that Tr(AB) = Tr( BA) using index notation. Hint: first write out the matrix multiplication AB in index notation, then contract the upper and lower indices to take the trace, and finally rearrange to find the contraction corresponding to BA. (b) Show that Tr(ABC) = Tr(BCA) = Tr(CAB). Hint: use the result from part (a) and the associativity of matrix multiplication to group the product into 2 terms. (c) Show that the result from part (b) generalizes to an arbitrary number of terms in the product: Tr(AjA2 . . . An-14,) = Tr(An AjA2 . .. An-1). This is called the cyclic property of the trace because the trace remains the same after any matrix is cycled from the end of a product of matrices to the beginning. 2. The orthogonal group. (40 points) The set of n x n orthogonal matrices M can be defined as those satisfying MM = I, where I is the n x n identity matrix. Show that this set of matrices forms a group for any n, using only the defining relationship MTM = / and the rules for matrix multiplication with transposes and inverses as discussed in lecture. This group is called the orthogonal group O(n). Hint: the goal is to show that the identity element, the inverse element, and the product of two group elements all satisfy the defining relationship, thus establishing that they belong to the group. In these calculations, you will probably find that matrix notation, rather than index notation, is more convenient. 3. Affine transformations form a group. (30 points) Show that the affine transformations from HW 5, problem 3 form a group. Hint: be careful about how you define the inverse element! The ordered-pair (R(0), a) representation is easiest for this purpose, rather than the 3 x 3 matrix representation.3. Matrix multiplication and affine transformations. In week 3 you saw that the matrix cos d - sin d MA = sin e cos d transformed the first two components of a vector by rotating 0 1 it through an angle # and adding the vector a = (ro, yo). Another way to represent this cos e - sin e transformation is an ordered pair A = (R(8), a), where R(0) = is the 2 x 2 sin d cos B matrix in the upper-left corner of the 3 x 3 matrix MA. This is an affine transformation which acts on a 2-component vector ~ and returns w = R(0)7 + a. (a) (15 points) If we apply a second affine transformation B = (R(d), b) to w, what is the resulting 2-component vector # in terms of R(8), R(d), a, b, and ? (b) (15 points) If we apply the transformations of part (a) in the opposite order (B applied to ", then A applied to the result), what is the resulting two-component vector? You should find that it is not the same an the answer from part (a). Explain this difference geometrically (using a sketch if you find it helpful). (c) (20 points) Letting 6 = (21, 31), write B as a 3 x 3 matrix My in the same form as MA. Compute the matrix products MAMA and MAMe, and explain how applying these matrices to V = (21, 12, 1) agrees with your computations from (a) and (b)