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Spring 2012, Math 54 GSI: Shishir Agrawal Discussions 106 and 107 Worksheet 5 Problem 1. Let P2 be the vector space of polynomials of degree at most 2. Let S standard basis for P2 , and let B t1, 1 t, 1 2t 2t2 u. t1, t, t2 u be the (a) Show that B is a basis for P2 . (b) Calculate the change-of-basis matrices PBÐS and PSÐB . (c) Find the B-coordinate vector of the polynomial 2 2t2 . 4t Problem 2. Let M2 be the set of all 2 2 matrices. (a) Show that M2 is a vector space (under matrix addition and scalar multiplication). (b) What is the dimension of M2 ? Give a basis for M2 . 1 2 and then let V be the the set of all 2 2 matrices A such that AB 2 4 is a subspace of M2 . (c) Let B 0. Show that V (d) What is the dimension of V ? Find a basis for V . Solution. For part (b), observe that " 1 0 is a basis for M2 , so dim M 2 4. a b For part (d), let A . Then AB c d if and only if a only if a c 2b 0 and c b d 0 0 , 0 0 0 0 , 0 0 * 0 1 0 if and only if 1 2 1 0 , 0 1 2 4 a c 2b 2d 2a 2c 4b 4d 0 0 0 0 2d 0, if and only if a 2b and c 2d. In other words, AB A 2b 2d b d b 02 1 0 c 0 2 0 1 for arbitrary real numbers b and c. Thus V is spanned by the set " 2 0 1 , 0 0 2 * 0 1 which are clearly linearly independent. Thus this is a basis and dim V Problem 3. 1 2 Let A 0 1 1 1 0 2. 0 (a) Find the characteristic polynomial of A. 1 2. 0 if and (b) Find the eigenvalues of A. (c) Is A diagonalizable? Problem 4. Determine if each of the following statements are true or false. (a) Every square matrix has at least 1 real eigenvalue. (b) Every 3 3 matrix has at least 1 real eigenvalue. (c) The sum of two eigenvalues of a matrix A is an eigenvalue of A. (d) The sum of two eigenvectors of a matrix A is an eigenvector of A. Solution. False: polynomials of even degree may not have real roots. True: cubic polynomials must have a real root by the intermediate value theorem. False: almost anything is a counterexample. False: take two eigenvectors corresponding to different eigenvalues to get a counterexample. Problem 5. Let C 8 be the vector space of all infinitely differentiable functions f : R Ñ R, and let T : C 8 Ñ C 8 be the linear transformation defined by T pf q f 1 . Using a bit of calculus, find the eigenvalues and associated eigenspaces of T . Solution. A real number λ is an eigenvalue of T if and only if there is a nonzero function f such that f 1 λf , if and only if f 1 pxq{f pxq λ, if and only if log f pxq λx d for some constant d, if and only if f pxq ceλx for some constant c. In other words, every real number λ is an eigenvalue of T , and the eigenspace associated to λ is the one-dimensional subspace of C 8 spanned by the infinitely differentiable function f pxq eλx . 2