The University of New South Wales School of Mathematics and Statistics MATH2501 LINEAR ALGEBRA Session 1, 2016 SAMPLE TEST 2 Student's surname Given name or initials Student number Time allowed: 45 minutes. Question 1 (3 marks) Write, as usual, X T for the transpose of a matrix X, and define a function 2 F : M2,2 R , where \u0012 \u0013 \u0012 \u0013 1 3 T F (X) = X +X . 2 4 Prove that F is a linear transformation. Be sure to set out your argument clearly and logically. Question 2 (4 marks) Find the line y = a + bx which best fits in the least squares sense the points (1, 1) , Question 3 (1, 6) , (2, 1) , (4, 7) . (6 marks) (a) Find an orthonormal basis for the span of the vectors x1 = (2, 3, 1, 1) , x2 = (1, 5, 3, 1) , x3 = (2, 6, 7, 8) . (b) Find a QR factorisation of the matrix 2 3 A= 1 1 Question 4 1 2 5 6 . 3 7 1 8 (3 marks) Find the reflection of the vector v = (4, 7, 1) in the plane 5x1 x2 + 2x3 = 0. Question 5 (4 marks) (a) Find the eigenvalues and eigenvectors of the matrix B = (b) Is B diagonalisable? Give reasons. \u0012 7 9 \u0013 1 . 1 Question 6 Check one of your answers to a previous question. Up to 2 bonus marks will be given for a good check. Make sure your working is clearly labelled \"Question 6\