THE UNIVERSITY OF NEW SOUTH WALES SCHOOL OF MATHEMATICS AND STATISTICS JUNE 2016 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 2016 MATH2501 Page 2 START A NEW BOOKLET 1. a) Let 1 3 0 3 1 1 1 1 A= 0 4 2 8 . 2 0 3 1 i) Find, giving reasons, the nullity and the rank of A. ii) Find a basis for the kernel (nullspace) of A. iii) If x0 is a given vector in R4 and y = Ax0 , write down the general solution of Ax = y. iv) Is there a vector b in R4 such that Ax = b has no solution? If so, give an example of such a vector; if not, explain why not. b) Find the line y = a + bx which best fits in the least squares sense the points (2, 3) , (3, 2) , (5, 1) , and (6, 0) . c) Let x1 = (1, 3, 1, 1) , x2 = (6, 8, 2, 4) and x3 = (6, 3, 6, 3) , and write W = span{ x1 , x2 , x3 }. i) Find an orthonormal basis for W . ii) Find a QR factorisation of 1 6 6 3 8 3 B= 1 2 6 . 1 4 3 iii) Show that y = (1, 1, 1, 3) is in W . iv) Find an orthonormal basis for R4 which includes the basis you found in (i). Give reasons. Please see over . . . JUNE 2016 2. MATH2501 Page 3 START A NEW BOOKLET a) In solving the following problems, give clearly written and logically complete arguments. i) Suppose that V and W are two subspaces of Rn for some positive integer n. Prove that V W is also a subspace of Rn . ii) Is V W is subspace of Rn ? Why or why not? iii) Prove further that W W = {0}. That is, the only vector that a subspace and its orthogonal complement have in common is the zero vector. b) Let V = P4 and W = {p(x) V : p(1) = p (1) = 0}. i) Assume that W is a subspace of V . Find a basis for W and thereby determine the dimension of W . ii) Given f and g in V , show that 1 < f, g >= f (t)g(t)dt 0 defines an inner product on V . c) Let v1 = (1, 2, 1) , v2 = (2, 1, 1) , and let W be the plane spanned by v1 and v2 . Consider the function T : R3 R3 where T (x) = projW x , that is, T (x) is the projection of x onto the plane W ; you may assume that T is a linear transformation. i) Evaluate T (4, 0, 9). ii) Find a basis for W in R3 . iii) Without calculation, write down the matrix of T with respect to the ordered basis { v1 , v2 , v3 }, where v3 is the basis element found in part (ii). By drawing a diagram, or otherwise, give reasons for your answer. iv) Hence, or otherwise, find an expression for the matrix of T with respect to the standard basis in R3 . You may leave your answer as a product of matrices without completing the calculation. Please see over . . . JUNE 2016 MATH2501 Page 4 START A NEW BOOKLET 3. a) Let 2 9 6 1 9 2 6 . Q= 11 6 6 7 It is given that Q has determinant 1. i) Show that Q is an orthogonal matrix. ii) Without further calcululations, describe the geometric eect of multiplying a non-zero vector by Q. iii) By finding the angle of rotation, show that Q is, in fact, purely a reflection matrix. iv) Find, in cartesian form, the plane of reflection. v) 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 conic 3x2 4xy + 6y 2 = 1. i) Express the conic in matrix form xT Ax = 1, where A is a symmetric matrix. ii) Find the eigenvalues and eigenvectors of A. iii) By making an appropriate change of variable, express the conic in standard form and classify it. iv) Make a neat large sketch of the conic showing both the (x, y) and principal axes. v) Find the (x, y) coordinates of the point(s) on the conic closest to the origin. c) A matrix B has characteristic polynomial p(x) = x3 2x + 1. i) Explain why B has an inverse. ii) Express B 1 as a quadratic in B. d) Find a matrix X, in terms of B, such that (B I)X = 0, where 0 is the zero matrix. Please see over . . . JUNE 2016 MATH2501 Page 5 START A NEW BOOKLET 4. a) Let i) ii) iii) iv) 0 1 0 1. A = 0 1 0 1 1 Find A2 and A3 . Explain why A has sole eigenvalue 0. Find a Jordan form for A. Find a matrix P such that P 1 AP is in Jordan form. b) Let 1 1 0 2 4 3 3 4 , v1 = 1 , v2 = 0 , v3 = 1 . B= 4 1 1 1 3 4 4 i) Find Bv1 , Bv2 and deduce that 1, 3 are the eigenvalues of B. ii) Find (B + I)v3 and hence calculate eBt v3 . c) Let ) ( 1 1 . C= 4 3 i) Show that 1 is the only eigenvalue of C. ( ii) Solve the system y = Cy + b(t), where b(t) = e ( ) 1 to y(0) = . 2 t 1 2 ) , subject Please see over . . . JUNE 2016 5. MATH2501 Page 6 START A NEW BOOKLET a) A linear transformation P : V V is said to be idempotent if P (P (v)) = P (v) for all v V. The projection map is an example of an idempotent linear transformation. i) Show that the only possible eigenvalues for an idempotent linear transformation are 0 and 1. ii) Show that if P is idempotent and P is neither the zero nor the identity transformation on V , then both 0 and 1 are eigenvalues. b) A 33 matrixA has 0.438, eigenvalues 1, 0.123, with corresponding eigen1 2 1 2 , v2 = 4 2 . vectors v1 = , v3 = 2 6 5 Let M be the limit of the sequence of matrices A, A2 , A3 , .... i) Find A3 v1 and write down an expression for A3 v2 . ii) By noting that the eigenvectors form a basis for R3 , or otherwise, find a basis for the column space of M and state the rank of M . iii) Find a basis for the kernel of M and state the nullity. c) Let v be a unit vector in Rn , and let Q = I 2vvT . i) Prove that Q is a symmetric matrix and also an orthogonal matrix. ii) Find Qv. iii) Deduce that det(Q2 I) = 0