THE UNIVERSITY OF NEW SOUTH WALES SCHOOL OF MATHEMATICS AND STATISTICS JUNE 2015 MATH2501 Linear Algebra (1) TIME ALLOWED - THREE HOURS. (2) TOTAL NUMBER OF QUESTIONS - 5 (3) ANSWER ALL QUESTIONS (4) THE QUESTIONS ARE OF EQUAL VALUE (5) THIS PAPER MAY BE RETAINED BY THE CANDIDATE (6) ONLY CALCULATORS BEARING A \"UNSW APPROVED\" STICKER MAY BE USED. (7) TO OBTAIN FULL MARKS, YOUR ANSWERS MUST NOT ONLY BE CORRECT, BUT ALSO ADEQUATELY EXPLAINED, CLEARLY WRITTEN AND LOGICALLY SET OUT. All answers must be written in ink. Except where they are expressly required pencils may only be used for drawing, sketching or graphical work. JUNE 2015 1. a) Let MATH2501 Page 2 1 2 1 2 3 5 0 7 A= 1 4 8 2 . 2 3 1 5 i) ii) iii) iv) Find a basis for the column space of A. Find a basis for the nullspace (kernel) of A. Write down the rank and the nullity of A. If x0 and y are vectors in R4 such that Ax0 = y, write down the general solution of the system Ax = y. v) Is there any vector z in R4 for which the system Ax = z has no solution? Answer YES or NO, and give reasons. b) A parabola whose axis is the y-axis has equation y = a+bx2 . It is desired to find the curve of this form which best fits, in the least squares sense, the points (1, 7) , (0, 1) , (1, 3) , (2, 5) . i) Write in matrix form the equations which a and b must satisfy if the curve is to go exactly through the given points. ii) Hence find the curve which best fits the points. c) Let W = span{ (1, 2, 3, 4), (2, 5, 0, 1) }. i) Find the projection onto W of v = (6, 3, 8, 5). ii) Find a basis for the orthogonal complement of W in R4 . Please see over . . . JUNE 2015 2. MATH2501 Page 3 a) In solving the following problems, give clearly written and logically complete arguments; and make sure that the dierences between the two problems are clear. In both problems, A is a fixed 2 2 matrix, and X T denotes the transpose of X. i) Prove that S = { X M2,2 | AX X T = 0 } is a vector space. ii) Prove that the function f : M2,2 M2,2 where f (X) = AX X T is a linear transformation. b) Let b1 = (1, 2, 5), b2 = (1, 4, 1), b3 = (4, 1, 8) and v = (9, 1, 3). i) Prove that B = { b1 , b2 , b3 } is a basis for R3 . ii) Find the coordinate vector of v with respect to the ordered basis B = { b1 , b2 , b3 }. c) Let a be a real number. It is given that the set C = { eax , xeax , x2 eax } is linearly independent; let V = span(C). Define a linear transformation T :V V where T (f ) = df . dx i) Find the matrix of T with respect to the ordered basis C for V . ii) Use the matrix you obtained in (i) to find the second derivative of x2 eax . Note: no marks will be given for an answer obtained by any other method. d) For the matrix ( ) 4 1 A= , 6 1 find a formula for An , giving an explicit expression for each entry in your answer. Please see over . . . JUNE 2015 3. a) Let MATH2501 Page 4 1 2 2 1 Q = 2 1 2 . 3 2 2 1 It is given that Q has determinant 1. i) Show that Q is an orthogonal matrix. ii) Explain without calculation how you know that Q is a rotation matrix, and find the angle of rotation. iii) Find the equation of the axis of the rotation. iv) Without doing any further calculations, find a formula for Qn , where n is a positive integer. You will need to consider two cases. b) Consider the surface in R3 6x2 y 2 + 2z 2 + 4xy 2xz + 4yz = 12. This equation can be written as xT Ax = 12 where x 6 2 1 2 1 2 and x = y . A= z 1 2 2 You are given that = 7 and = 1 are two of the eigenvalues of A, with corresponding eigenvectors 5 1 2 and 2 1 1 respectively. i) Find the third eigenvalue and its corresponding eigenvector. ii) Hence write down an orthogonal matrix P , such that the change of variables (x, y, z)T = P (X, Y, Z)T puts the surface into standard form, and give the standard form in terms of the principal axes X, Y, Z. iii) What is the shortest distance from the surface to the origin? iv) Find the (x, y, z) coordinates of the closest points on the surface to the origin. c) It is given that the matrix 5 6 6 2 A = 1 4 3 6 4 has characteristic polynomia p() = 3 52 + 8 4 = ( 1)( 2)2 . i) State the Cayley-Hamilton Theorem. ii) Express A4 as a quadratic in A. iii) Find the minimal polynomial for A. Give reasons. Please see over . . . JUNE 2015 4. a) Let MATH2501 Page 5 5 1 4 4 1 4 A= 2 2 5 1 0 1 0 0 . 0 3 The matrix A has one eigenvalue = 3, with 0 1 2 0 ker(A 3I) = span , , 0 1 1 2 1 0 1 2 0 , , 0 . ker(A 3I)2 = span 1 0 0 2 1 0 i) Write down a Jordan form for A. ii) Find a chain of length 3 based on the generalised eigenvectors for A. iii) Hence find an invertible matrix P such that P 1 AP is in Jordan form. b) A 7 7 matrix D has sole eigenvalue = 5. You are given that dim(ker(D 5I)) = 3 and dim(ker(D 5I)2 ) = 5 . Find the possible forms for the Jordan form for D. ( ) 1 4 c) Let B = . 1 5 i) Show that B is not diagonalisable. ii) Find (B 3I)2 . iii) Hence, or otherwise, find eBt . d) Solve the non-homogeneous system of dierential equations dy1 = y1 + 4y2 + 2t dt dy2 = y1 + 5y2 + t dt subject to initial conditions y1 (0) = 0, y2 (0) = 0. You may use any results from your calculations in (c) above. Please see over . . . JUNE 2015 5. MATH2501 Page 6 a) Let S be the vector space of symmetric 3 3 matrices X which satisfy the condition 1 0 X 2 = 0 . 3 0 Find a basis for S, and state the dimension of S. b) Consider the function T : R2 R2 , where T (x) is the reflection of the vector x in the line y = mx. i) Explain why ( ) 1 v1 = m ( and v2 = ) m 1 are eigenvectors for T ; and state their corresponding eigenvalues. ii) Hence write down the matrix representation for T with respect to the ordered basis B = {v1 , v2 } in both domain and codomain. iii) By using a change of basis argument, or otherwise, find the matrix for T with respect to the standard basis in both domain and codomain. c) Let n be a positive integer, let W be a subspace of Rn and let W be the orthogonal complement of W in Rn . Prove that the only vector which lies in both W and W is the zero vector. d) Let u and v be two linearly independent vectors in R2 . i) Prove that (u u)(v v) (u v)2 > 0. ii) Prove that xu + yv = 1 is the equation of an ellipse