November 2018 MATH2099 Page 2 Answer question in a separate book marked Question 1 1. [20 marks] i) [4 marks] Let P2(R) be the vector space of all real polynomials of degree less than or equal to 2 with ordered basis B = {x2, 1− x, 1 + x}. a) A polynomial u ∈ P2(R) has coordinate vector [u]B = ⎛ ⎝ 31 2 ⎞ ⎠ with respect to B. Find u. b) Find the coordinate vector [v]B of v = 3x2 + 4x + 6 ∈ P2(R) with respect to the ordered basis B. ii) [4 marks] Find the parabola of the form y = αx2 + β which best fits the points {( 0 0 ) , ( 2 2 ) , ( 1 2 ) , ( 1 0 )} in the least squares sense. iii) [4 marks] The linear transformation T reflects R3 in the y − z plane. a) Find the standard matrix A of T . b) Write down (A−1)2018. iv) [8 marks] Let v1 = ⎛ ⎝ 10 0 ⎞ ⎠ and v2 = ⎛ ⎝ 21 2 ⎞ ⎠ be two vectors in R3 and let W = span{v1,v2}. a) Use the Gram-Schmidt process to find an orthonormal basis for W . b) Using the orthonormal basis from a), show that the projection of the vector z = ⎛ ⎝ 510 5 ⎞ ⎠ onto W is equal to ⎛ ⎝ 54 8 ⎞ ⎠. c) Express z as a sum, z = w1 +w2 where w1 ∈ W and w2 ∈ W⊥. d) Find a vector y in R3, different from z, with the property that ProjW (y) = ProjW (z) and ∥y∥ = ∥z∥. Please see over . . . November 2018 MATH2099 Page 3 Answer question in a separate book marked Question 3 2. [20 marks] i) [3 marks] Suppose that V is an n dimensional real vector space, W is an m dimensional real vector space and that T : V → W is linear. Exactly 3 out of the following 12 statements are true for all V , W and T . Write down the numbers of these three true statements. 1) Rank(V ) + Nullity(W ) = Dim(T ). 2) Rank(T ) + Nullity(T ) = Dim(V ). 3) Rank(W ) + Nullity(V ) = Dim(T ). 4) Rank(T ) + Nullity(T ) = Dim(W ). 5) A spanning set for V must contain exactly n vectors. 6) A spanning set for V must contain exactly m vectors. 7) A spanning set for V must contain at least n vectors. 8) A spanning set for V must contain at most n vectors. 9) T (αv1 + v2) = αT (v1) + T (v2) for all v1,v2 ∈ V and α ∈ R. 10) T (αv1 + v2) = T (v1 + αv2) for all v1,v2 ∈ V and α ∈ R. 11) T (αv1 + v2) = αT (v1 + v2) for all v1,v2 ∈ V and α ∈ R. 12) T (αv1 + v2) = T (v1) + αT (v2) for all v1,v2 ∈ V and α ∈ R. ii) [3 marks] Suppose that A is an 8 × 8 matrix with a single eigenvalue λ = 3. You are given that Ker(A−3I) is 2-dimensional, Ker(A−3I)2 is 4- dimensional, Ker(A−3I)3 is 6-dimensional, Ker(A−3I)4 is 7-dimensional and Ker(A − 3I)5 is 8-dimensional. By considering an appropriate ring diagram, find a Jordan matrix J similar to A. Please see over . . . November 2018 MATH2099 Page 4 iii) [4 marks] Consider the following commutative diagram for a linear transformation T : R2 → R2. A F = ( 3 4 5 7 ) B = {( 1 0 ) , ( 0 1 )} R2 R2 B = {( 1 0 ) , ( 0 1 )} P Q D = {( 2 1 ) , ( 1 1 )} R2 R2 B = {( 1 0 ) , ( 0 1 )} Find the standard matrix A of T . iv) [7 marks] Let G = ( 5 −4 1 1 ) . a) By finding all the eigenvalues and eigenvectors of G show that G is not diagonalisable. b) Find an invertible matrix P and a Jordan matrix J such that P−1GP = J . c) Calculate eGt. v) [3 marks] Suppose that u, v ∈ Rn and that u ·v = 5. Define the matrix C to be C = I − AB where A = uuT , B = vvT , and I is the n×n identity matrix. Prove that u is an eigenvector of C and find the associated eigenvalue. December 2018 Additional Assessment MATH2859 Page 2 1. Answer in a separate book marked Question 1 i) Let X follow the Bernoulli distribution: p(x) = ⇢ 1 ⇡, if x = 0 ⇡, if x = 1 where 0 < ⇡ < 1. a) [1 mark] Show that E(X) = ⇡. b) [2 marks] Show that Var(X) = ⇡(1 ⇡). c) [3 marks] Assume we have a random sample {X1, . . . , Xn} of size n from X. The probability ⇡ can be estimated from this sample as Pˆ = 1 n nX i=1 Xi. Show that the standard error of Pˆ is equal tor ⇡(1 ⇡) n . Make sure you include your reasoning. ii) (Matlab output relevant to this question can be found at the end of the question.) In August this year, Roy Morgan Research published a poll of New Zealanders on television viewing habits related to sport. The poll of 6,422 randomly selected New Zealanders found that 43.6% of them watch rugby on the television. a) [3 marks] Find a 99% confidence interval for the true proportion of New Zealanders who watch rugby on the television. b) [3 marks] What assumptions did you make in the above? Where possible, check if these assumptions are reasonable. c) [2 marks] Let’s say Roy Morgan Research wanted to estimate the proportion of rugby viewers to within 1% of its true value (with 95% confidence). How many people did they need to sample to achieve this? Please see over . . . December 2018 Additional Assessment MATH2859 Page 3 iii) (Matlab output relevant to this question can be found at the end of the question.) Assume Rugby New Zealand (the organising body for the sport) want to be able to demonstrate that the proportion of New Zealanders watching rugby is in excess of 40% , using a new sample of size n (to be determined). They want to do this by a suitable hypothesis test. a) [1 mark] What are the appropriate null and alternative hypotheses for this test? b) [1 mark] What is the approximate distribution of the sample pro- portion Pˆ ,if the null hypothesis is true? Please write your answer as a function of n. c) [2 marks] Show that, for the relevant hypothesis test at the 0.01 significance level, the rejection region can be expressed as pˆ 2 0.4 + 1.139p n , 1