ENSC 801 Assignment 3,4 November 13, 2017 1. Create an orthonormal basis starting with basis elements 1, t, t2 in L2 [1, 1]. 2. David Luenberger text, Chapter 3, problem 5 3. David Luenberger text, Chapter 3, problem 10 4. David Luenberger text, Chapter 3, problem 17 5. State the projection theorem. 6. Consider that an experiment gave three scalar data points y to be 3, 5, 8 at times t = 1, 2, 3 respectively. The experiment model is linear y = c + mt. Find the best line (parameters c, m) that fits the data. 7. Consider that an experiment gave four scalar data points y to be 1, 3, 7, 11 at times t = 1, 2, 3, 4 respectively. The experiment model is y = c + mt + nt2 . Find the best curve (parameters c, m, n) that fits the data. 8. State the modified projection theorem. 9. Characterize all the solutions of this system of equations (scalars 1 , 2 given): x1 + x2 + x3 + x4 + x5 + x6 + x7 = 3x1 + 4x2 + 5x3 + 4x4 + 3x5 + 2x6 + x7 = . (1) Hint: use the modified projection theorem. 10. The optimal decision boundary in many pattern discrimination problems is of the form: w t x + w0 = 0 where x, w Rn , w0 is a scalar, w, w0 are given and x is unknown. Show that the best x is given by: x= w0 w. kwk2 Hint: Use the modified projection theorem. 1 11. Consider that an experiment gave four scalar data points y to be 1, 3, 7, 11 at times t = 1, 2, 3, 4 respectively. The experiment model is y = c + mt + nt2 + rt3 . Find the best curve (parameters c, m, n, r) that fits the data. 12. The well-known Rayleigh quotient, which also occurs in several pattern matching problems, is given by: J(x) = xt P x xt Qx where P, Q are matrices in Rnn . The task is to find the unknown vector x Rn that maximizes J. Show that the solution must satisfy the generalized eigen-value problem P x = Qx. Hint: Use Lagrange Multipliers and calculus. 2