MATH 2350 Assignment 3 Due Wednesday March 8th at 16:00 Problem 1: Use Gram Schmidt to transform the following bases into orthonormal bases. 1 0 1 i) {0 ; 1 ; 3} . 1 1 3 1 1 1 0 2 1 2 0 ii) { 1 ; 0 ; 0 ; 0} . 1 1 0 0 Problem 2: Find the QR factorisation 1 1 A = 1 0 0 1 Problem 3: Orthogonally diagonalize 2 3 A = 3 2 0 4 of 0 1 1 0 4 2 Hint: The eigenvalues are 3, 2, 7. Problem 4: Find a symmetric 2 2 matrix with eigenvalues = 3, = 1 and corresponding eigenvectors \u0014 \u0015 \u0014 \u0015 1 2 v1 = ; v2 = 2 1 Problem 5: Find the symmetric matrices which give the following quadratic forms: (i) Q = x22 2x23 + 2x1 x3 + x2 x3 . (ii) Q = 3x21 + 2x22 + x23 3x1 x4 + x3 x4 . Problem 6: Consider the quadratic form Q = x2 + 8xy + y 2 1 (i) Find the change of variable x = P y which eliminates the cross term from Q. Write Q in the new form. (ii) Is Q positive definite, positive semi-definite, negative definite, negative semi-definite or indefinite? Problem 7: Identify the graph of the conic 3x21 + 2x1 x2 + 3x22 = 8 . Sketch the curve. Problem 8: Let A be a square matrix and let A = QR be the QRfactorisation of A. Prove that A is invertible if and only if R has no zero on the diagonal. 2