- June 13, 2020

Student Number Semester 2 Assessment, 2019 School of Mathematics and Statistics MAST10007 Linear Algebra Writing time: 3 hours Reading time: 15 minutes This is NOT an open book exam This paper consists of 6 pages (including this page) Authorised Materials • Mobile phones, smart watches and internet or communication devices are forbidden. • No written or printed materials may be brought into the examination. • No calculators of any kind may be brought into the examination. Instructions to Students • You must NOT remove this question paper at the conclusion of the examination. • There are 13 questions on this exam paper. • All questions may be attempted. • Start each question on a new page. Clearly label each page with the number of the question that you are attempting. • Marks may be awarded for – Using appropriate mathematical techniques. – Accuracy of the solution. – Full explanations, including justification of rules or theorems used. – Using correct mathematical notation. • The total number of marks available is 120. Instructions to Invigilators • Students must NOT remove this question paper at the conclusion of the examination. • Initially students are to receive the exam paper and two 11 page script books. This paper may be held in the Baillieu Library MAST10007 Semester 2, 2019 Question 1 (9 marks) Consider the system of equations x + 2y + 2z = 2 2x + 5y + 3z = 5 x + 3y + k2z = k + 2 where x, y, z ∈ R and k ∈ R. (a) Determine the values of k, if any, for which the system has (i) a unique solution, (ii) no solutions, (iii) infinitely many solutions. (b) Find all solutions to the system when k = 1. Question 2 (11 marks) (a) Consider the matrices A = [ 1 2 3 5 ] , B = [ 2 0 −1 1 −2 4 ] , C = [ 1 2 3 0 5 −1 ] . Calculate the following, if they exist: (i) AB, (ii) BCT . (b) Prove that if A and B are matrices such that AB and BA are both defined, then AB and BA are both square matrices. (c) Use cofactor expansion to find the determinant of the matrix A = a 0 b 0 0 a 0 b c 0 d 0 0 c 0 d , where a, b, c, d are complex numbers. When is A invertible? Explain your answer. Question 3 (5 marks) Let A = 0 1 11 0 1 1 1 0 . (a) Verify that A2 −A = 2I, where I is the 3× 3 identity matrix. (b) Deduce from part (a) that A is invertible, and that A−1 = 12(A− I). Page 2 of 6 pages MAST10007 Semester 2, 2019 Question 4 (12 marks) Consider the points P (1, 1, 0), Q(0, 1, 2) and R(1, 0, 1) in R3. (a) Find the distance between P and Q. (b) Find the cosine of the angle between the vectors −→ PQ and −→ PR. (c) Find the area of the triangle with vertices P , Q and R. (d) Find a Cartesian equation for the plane that passes through P , Q and R. Question 5 (10 marks) In each part of this question, determine whether W is a subspace of the real vector space V . For each part, give a complete proof using the subspace theorem, or a specific counterexample to show that some subspace property fails. (a) V = R4, W = {(a, b, c, d) ∈ R4 | a+ b+ c+ d = 1}. (b) V = P3, W = {p(x) ∈ P3 | p(x) = p(−x) for all x ∈ R}. (c) V = M2,2, W = {A ∈M2,2 | A2 = A}. Question 6 (9 marks) The following two matrices are related by a sequence of elementary row operations: A = 1 2 −1 0 2 1 4 3 1 0 3 2 2 −1 8 5 , B = 1 0 3 2 0 1 −2 −1 0 0 0 0 0 0 0 0 Let W be the subspace of R4 spanned by the set of vectors S = {(1, 2, 1, 2), (2, 1, 0,−1), (−1, 4, 3, 8), (0, 3, 2, 5)}. (a) Find a subset of S that is a basis for W . Hence, find the dimension of W . (b) Write the other vectors in S as a linear combination of your basis vectors. (c) Consider the set of vectors T = {(−1, 4, 3, 8), (0, 3, 2, 5)}. Is T linearly independent? Is T a basis for W? Explain your answers. Page 3 of 6 pages MAST10007 Semester 2, 2019 Question 7 (11 marks) Let T : R3 → R3 be the linear transformation given by T (x, y, z) = (x+ y + 2z, y + 2z, z). Consider the bases of R3 given by S = {(1, 0, 0), (0, 1, 0), (0, 0, 1)} and B = {(1, 0, 0), (1,−1, 0), (2,−1,−1)} . (a) Find the matrix [T ]S of T with respect to the standard basis S. (b) Is T (i) injective, (ii) surjective, (iii) invertible? Explain your answers. (c) Find the transition matrix PS,B. (d) Find the transition matrix PB,S . (e) Find the matrix [T ]B of T with respect to the basis B. Question 8 (13 marks) Consider the function T : M2,2 →M2,2 given by T (X) = [ 2 2 1 1 ] X. (a) Show that T is a linear transformation. (b) Find the matrix [T ]S of T with respect to the standard basis S = {[ 1 0 0 0 ] , [ 0 1 0 0 ] , [ 0 0 1 0 ] , [ 0 0 0 1 ]} (c) Find a basis for the kernel of T . (d) Find a basis for the image of T . (e) Verify the rank-nullity theorem for the linear transformation T . Question 9 (6 marks) For each of the following matrices, determine whether it is diagonalisable and give a short justification: A = [ i 1 + i 0 i ] , B = −2 5 10 3 −6 0 0 −4 , C = −2 5 15 −2 −6 1 −6 −2 . (Hint: Very little calculation should be needed to answer this question.) Page 4 of 6 pages MAST10007 Semester 2, 2019 Question 10 (13 marks) In a certain town, the weather each day is either rainy or fine. • If the weather is rainy one day, then it is rainy the next day 60% of the time. • If the weather is fine one day, then it is fine the next day 80% of the time. Let rn be the probability that the weather is rainy after n days, and fn be the probability that the weather is fine after n days. (a) Explain briefly why [ rn+1 fn+1 ] = A [ rn fn ] , where A = [ 0.6 0.2 0.4 0.8 ] . (b) Find the eigenvalues and corresponding eigenvectors for A. (c) Find an invertible matrix P and a diagonal matrix D such that A = PDP−1. (d) Assuming that today is fine we have r0 = 0 and f0 = 1. Find formulas for rn and fn for n ≥ 1. (e) What are the long term probabilities of rainy days rn and fine days fn, as n→∞? Question 11 (10 marks) Consider R3 with the standard inner product given by the dot product 〈u,v〉 = u · v = u1v1 + u2v2 + u3v3. Let W ⊂ R3 be the subspace spanned by {(0, 1, 1), (1, 0, 1)}. (a) Find an orthonormal basis for W . (b) For v = (1, 1, 0) ∈ R3, find (i) the orthogonal projection of v onto W , (ii) the distance from v to W . Page 5 of 6 pages MAST10007 Semester 2, 2019 Question 12 (7 marks) (a) Find the least squares line of best fit y = a+ bx for the data points {(−1, 2), (0, 1), (1, 2), (2, 3)} (b) Draw a clear graph showing the data points and your line of best fit. Question 13 (4 marks) Let A be an n× n real matrix. Fix a real number λ and consider the set W = { w ∈ Rn | (A− λI)2w = 0} . Show that W 6= {0} if and only if λ is an eigenvalue of A. End of Exam—Total Available Marks = 120 Page 6 of 6 pages