School of Mathematics and Statistics University of New South Wales V T C J M12 T11 T14 W12 M13 T12 T15 W15 M15 MATH2501 Linear Algebra SESSION 2, 2019 TEST 2 Version A Student’s Surname Initials Student Number Questions: 4 Pages: 2 Total marks: 18 Time allowed: 40 minutes Q1 [5 marks] Find the projection of e1 ∈ R4 onto the subspace W = span 1 1 1 −1 , 1 −1 1 1 . Q2 [5 marks] Find QR factorisation of the matrix A = ( 5 17 12 7 ) . Q3 [3 marks] Let V = (V,+, ·,R) and W = (W,+, ·,R) be vector spaces and let T : V → W be a linear map. a) Give definition of the null space nullT of the map T . b) Let V = P2(R) and W = R2. Consider the subspace (you do not have to proof that it is a subspace). V = { p ∈ P2(R) : p(1) = 0 and p(−1) = 0 } ⊆ P2(R). Find a linear map T such that V = nullT. You do not have to show that the map T is linear. Q4 [5 marks] Let V = (V,+, ·,R) be a vector space and let B = {v1,v2,v3} ⊆ V be a basis. a) Define what it means that the triple (a1, a2, a3) ∈ R3 is the coordinate vector of x ∈ V with respect to the basis B. Page 1 of 2 School of Mathematics and Statistics University of New South Wales b) Find the coordinate vector of v1 with respect to the basis B. c) Define what it means that a matrix A ∈M3,3(R) is the matrix of the linear map T : V → V with respect to the basis B. d) Let A = 0 1 01 0 0 0 0 0 . be the matrix of the map T with respect to the basis B. Find T (v1) in terms of B. Page 2 of 2
