MATH 218 Final, Spring 2011

University of Wisconsin-Madison Math 340 - Spring 2010 Linear Algebra Final Exam Exercise 1: Let L : R2 → R2 such that L((1, 1)) = (3, 0) and L((−2, 1)) = (1, 1). Find L((1, 4)). Exercise 2: Some questions about the determinant. (1) Let A be a n × n matrix satisfying AAt = In. Prove that det A = 1 or det A = −1. (2) Show that there is no 5×5 matrices A satisfying At = −A (such a matrix is called skew symmetric). Exercise 3: Prove or disprove that the following maps are linear. (1) L1 : R2[X] → R with L1(a + bX + cX2) = a + b − 2 (2) L2 : Rn[X] → R, with L2(P) =< P, Q > where Q ∈ Rn[X] is fixed and < P, Q >= � 1 0 P(t)Q(t)dt. (3) L3 : R3 → R3, with L3((x, y, z)) = (x2, y − z, 0). Exercise 4: Let L : R2 → R2 be a linear map defined by L �� x y �� = � x + 2y x � Let C be the canonical basis and consider B = {v1, v2} of R2 with, v1 = � 1 −1 � , v2 = � 2 1 � . (1) Check that B is a basis of R2. (2) (a) Find the matrix of L relative to the canonical basis C, [L]C C. (b) Find the transition matrix, [I]C B, for changing from B to C. (c) Find the transition matrix, [I]B C , for changing from C to B. (d) Find the matrix of L relative to the basis B, [L]B B. (3) Consider the sequence of numbers un satisyfying u1 = 1, u2 = 1 and un+2 = un+1 + 2un. (we have 1, 1, 3, 5, 11, 21, 43...) (a) Find a matrix A such that � un+2 un+1 � = A � un+1 un � . (b) Find a relation between � un+2 un+1 � , A and � u2 u1 � . (c) compute u102. Exercise 5: Let A be an n × n matrix. The goal of this exercise is to prove that KerA = ImA if and only if 2rank(A) = n and A2 = 0. 1 2 (1) Assume that KerA = ImA. (a) Prove that 2rank(A) = n. (b) Prove that for all X ∈ Rn, A2X = 0. Deduce that A2 = 0. (2) Assume that 2rank(A) = n and A2 = 0. (a) Prove that ImA ⊆ KerA (b) Deduce that KerA = ImA by comparing their dimension. Exercise 6: Let L : R3[X] → R2[X] be the linear map defined by L(a + b + cX2 + dX3) = b − a + (c + d)X + aX2. Consider the basis C3 = {1, X, X2, X3} and C2 = {1, X, X2}. (1) Find the matrix [L]C2 C3 of L relative to C3 and C2. (2) Find a basis for kerL. (3) Find ImL. Exercise 7: Let (V, <, >) be an inner product space (of finite dimension) and let W be a subspace of V . Let projW : V → W be the orthogonal projection on W. (1) Prove that projW is linear (hint: take λ ∈ R, and v, v′ ∈ V and find the orthogonal decomposition of λv + v′). (2) Prove that Im projW = W. (3) Prove that ker projW = W ⊥ (hint: you can use the rank theorem).