MATH 218 Final, Fall 2010

University of Wisconsin-Madison Math 340 - Fall 2010 Linear Algebra Final Exam Exercise 1: (1) Let L1 : R3 → R3 be a linear map satisfying dimKerL1 = 3. Find L1. (2) Let L2 : R3[X] → R2[X] be a linear map satisfying dimKerL2 = 1. Find its image ImL2. (3) Is there any linear map L3 : R3 → R3[X] such that ImL3 = R3[X] ? Exercise 2: Let L : R2 → R2 such that L((−1, 1)) = (1, −3) and L((−1, 3)) = (0, 1). Find L((1, 1). Exercise 3: Prove or disprove that the following maps are linear. (1) L1 : R4[X] → R with L1(P) = 0. (2) L2 : R3 → R3, satisfying L((2, −1, 1)) = (1, 0, 1) and L((−2, 1, −1)) = (0, 1, 4). (3) L3 : R3 → R3, with L3((x, y, z)) = (x, xy, y + 2z). Exercise 4: Let L : R3 → R3 be a linear map defined by L     x y z     =   y 0 x + y   Let C be the canonical basis and consider B = {v1, v2, v3} of R3 with, v1 =   0 0 1   , v2 =   1 0 1   , v3 =   1 1 −1   . (1) Check that B is a basis of R3. (2) Find the matrix of L relative to the canonical basis C, [L]C C. (3) Find the transition matrix, [I]C B, for changing from B to C. (4) Find the transition matrix, [I]B C , for changing from C to B. (5) Find the matrix of L relative to the basis B, [L]B B. Exercise 5: Let M2×2(R) = �� a b c d � ; a, b, c, d ∈ R � the space of 2 × 2 matrices. Let L : M2×2(R) → M2×2(R) be the transformation defined by L(A) = At. (1) Show that L is linear. (2) Show that C = �� 1 0 0 0 � , � 0 1 0 0 � , � 0 0 1 0 � , � 0 0 0 1 �� is a basis of M2×2(R). (3) Find the matrix [L]C C of L relative to C. 1 2 Exercise 6: Consider the space W = span              1 √ 2 0 0 1 √ 2     ,     0 1 0 0     ,      1 √ 3 0 1 √ 3 −1 √ 3               . (1) Show that     1 √ 2 0 0 1 √ 2     ,     0 1 0 0     ,      1 √ 3 0 1 √ 3 −1 √ 3      is an orthonormal basis of W. (2) Compute projW         1 1 1 1        . Exercise 7: (1) Let L1 : R2[X] → R2[X] be the linear map defined by L(P) = P ′. (a) Compute L1 ◦ L1(a + bX + cX2). (b) Compute L1 ◦ L1 ◦ L1(a + bX + cX2). (2) Let L2 : R2[X] → R2[X] be the linear map defined by L2(a + bX + cX2) = c + aX + bX2. (a) Compute L2 ◦ L2(a + bX + cX2). (b) Compute L2 ◦ L2 ◦ L2(a + bX + cX2). (c) Compute L2 ◦ L2 ◦ · · · ◦ L2 � �� � n times (a + bX + cX2) for any n. Exercise 8: Consider the following inner product on R2[X]: < P, Q >:= � 1 −1 P(t)Q(t)dt. (1) Find a basis of {1}⊥ := {P ∈ R2[X], < P, 1 >= 0}. (2) What is the dimension of {1}⊥ ?