Financial Computing with Matlab Worksheet Sheet 4 Name This sheet will be marked and the marks gained used for assessment. Put your name in the box above and your answers in the boxes below. Hand it in to the school office by 1400 Thursday 8th December. 11. Consider the vectors a = 0@ 1 1 1 1A ; b = 0 @ 1 2 3 1 A ; b = 0 @ 1 2 1 1 A and x = 0 @ − 7 02 1 A : (a) Apply Gram{Schmidt to a; b; c to obtain orthonormal vectors e1; e2; e3. [12 marks] (b) Express the vector x in terms of e1; e2 and e3. [6 marks] 22. Consider the row vectors a = (a1; a2; a3); b = (b1; b2; b3); c = (c1; c2; c3): (a) Assume that a; b; c 2 R3 are linearly independent and let A = 0 @ a c b1 1 1 a c b2 2 2 a c b3 3 3 1 A : Show that the matrix AA0 is symmetric (where A0 is the transpose of A). [5 marks] (b) Assume that a; b; c 2 R3 are linearly independent. Show that AA0 is positive definite. [12 marks] 33. Consider the matrix A = 0BB@ 1 −1 0 0 −1 1 0 0 0 0 5 −1 0 0 −1 5 1CCA and the vector x = 0BB@ 1 2 3 4 1CCA : (a) Express x in terms of the eigenvectors of A. [5 marks] (b) Express Ax in terms of the eigenvectors and eigenvalues of A. [5 marks] (c) Express kAxk2 in terms of the eigenvectors and eigenvalues of A. [5 marks] 44. Consider the following vectors a = 0@ a1 a2 a3 1A ; b = 0 @ b b b1 2 3 1 A : (a) Calculate the vector c = b − proja(b): [5 marks] (b) Show that a and c are orthogonal [5 marks] (c) Set a = 0@ 1 3 1 1A ; b = 0 @ 1 2 1 1 A and verify that a is orthogonal to c = b − proja(b). [5 marks] 5