b (e) Let B = a C d be any 2 x 2 matrix. (i) Show that there are real numbers un and a such that Hint: express as a scalar multiple of a unit vector, and hence find an expression for un in terms of a and c. (ii) Let o E R. Use the invertibility of R. to prove that there are unique U12, 122 6 R such that COS OF - sin o = 112 + U22 sin a COS OF (ii) Use parts (i) and (ii) to show that B can be expressed in the form B = RU for some o E R and some upper-triangular matrix U. (iv) Suppose that B = RQU = RV, where o, B E R and U and V are upper- triangular. Prove that if B is invertible, then U = +V.(a) A rotation matrix is a 3 X 3 matrix R such that detfR] = 1 and R-1 : RT. Let R and S be rotation matrices1 and let t, u E R3. (i) Prove that RS is a rotation matrix. T11 T12 1r'13 a: R t (ii) Given R = r21 r22 1'23 and t. = y E R3, we write [0 1] for the 1"31 T32 7'33 3 4 x 4 matrix T11 1'12 T13 93 T21 T22 T23 29' T31 1'32 T33 2 0 I} 0 1 From page 1 of the attached paper, the product [3' i] [3 '1'] is a matrix of the form I: v]! where R' is a rotation matrix and v E R3. 0 1 Write down expressions for R and v in terms of R. S, t and n. For this question only, you do not need. to give any reasons for your answer. {1)} Suppose a robot is posed n times, leading to the equations 3,1ij = 31/33]. where RAJ. Rx, By and Raj are rotation matrices1 forj = 1, 2, . . .. n. Write a few sentences to summarise the results of the attached paper by Shah on: (i) the number of poses needed to obtain unique matrices RX and By which satisfy these equations; and (ii) how the position errors for the method presented in the attached paper oompare to the position errors for the method of Li et a], on simulated data and real-world data