2. Let A be a 3 x 3 matrix which represents the following transformation of R3: A reflection about the axis u = (1/13, -1/73,1/3)" (The transpose here is simply to denote that I mean to express u as a column vector). A rotation about the axis u = (1/V2,0,1/V2)" by an angle of /3. A reflection about the axis u = (0,1,07 You may assume that A has a determinant of 1, and that these transformations are applied sequentially, from the top to the bottom. (a) (5 marks) Let M, M1, M, be the matrices corresponding to the above trans- formations. Find each of these explicitly. You should give you matrices with exact (unrounded) entries. You may use the fact that a reflection matrix about the axis u is given by I 2uu? when u is a unit vector expressed as a column. (b) (5 marks) Find the matrix A. You may express your answer in terms of decimal values rounded to 4 decimal places. (c) (5 marks) We know that A is orthogonal and det(A) = 1, hence it represents a rotation about an axis u by some angle 6. Assuming that sin(0) = 0 and u has a positive first component, find 8 and u. Your answers may be given as decimal approximations, rounded to 4 decimal places. 2. Let A be a 3 x 3 matrix which represents the following transformation of R3: A reflection about the axis u = (1/13, -1/73,1/3)" (The transpose here is simply to denote that I mean to express u as a column vector). A rotation about the axis u = (1/V2,0,1/V2)" by an angle of /3. A reflection about the axis u = (0,1,07 You may assume that A has a determinant of 1, and that these transformations are applied sequentially, from the top to the bottom. (a) (5 marks) Let M, M1, M, be the matrices corresponding to the above trans- formations. Find each of these explicitly. You should give you matrices with exact (unrounded) entries. You may use the fact that a reflection matrix about the axis u is given by I 2uu? when u is a unit vector expressed as a column. (b) (5 marks) Find the matrix A. You may express your answer in terms of decimal values rounded to 4 decimal places. (c) (5 marks) We know that A is orthogonal and det(A) = 1, hence it represents a rotation about an axis u by some angle 6. Assuming that sin(0) = 0 and u has a positive first component, find 8 and u. Your answers may be given as decimal approximations, rounded to 4 decimal places