Math 110 Final Exam with Answers NAME 1 pt TA 1 pt Name of Neighbor to your left 1 pt Name of Neighbor to your right 1 pt Instructions You are allowed to use 1 sheet 8 1 2 by 11 inches both sides of notes Otherwise this is a closed book closed notes closed calculator closed computer closed PDA closed cellphone closed mp3 player closed network open brain exam You get one point each for lling in the 4 lines at the top of this page All other questions are worth 10 points Fill in the questions at the top of the page Then stop and wait until we tell you to turn the page and start the rest of the exam Do not start reading the rest of the exam until we tell you to start After you start read all the questions on the exam before you answer any of them so you do the ones you nd easier rst Write all your answers on this exam If you need scratch paper ask for it write your name on each sheet and attach it when you turn it in we have a stapler 1 2 3 4 5 6 7 8 Total 1 Question 1 10 points version 1 Let P R be the vector space of all real polynomials De ne the linear map T P R P R by T f f the second derivative Part 1 Show that T is onto but not one to one gi xi 2 so Answer If g x di 0 gi xi then T f g where f x f0 f1 x di 0 i 1 i 2 T is onto and f0 and f1 are arbitrary so T is not one to one Part 2 Describe all eigenvectors of T i e nonzero polynomials such that f x f x for some Answer Since the degree of f is less than the degree of f f cannot be a nonzero multiple of f Therefore 0 and f x 0 is satisfied by all linear polynomials f x f0 f1 x Question 1 10 points version 2 Let P C be the vector space of all complex polynomials De ne the linear map S P C P C by S g g the second derivative Part 1 Show that S is onto but not one to one fi xi 2 so Answer If f x di 0 fi xi then S g f where g x g0 g1 x di 0 i 1 i 2 S is onto and g0 and g1 are arbitrary so S is not one to one Part 2 Describe all eigenvectors of S i e nonzero polynomials such that g x g x for some Answer Since the degree of g is less than the degree of g g cannot be a nonzero multiple of g Therefore 0 and g x 0 is satisfied by all linear polynomials g x g0 g1 x 2 Question 2 10 points version 1 Let A be an m by n complex matrix and let B be an n by m complex matrix Show that Im A B is invertible if and only if In B A is invertible Answer Solution 1 Suppose Im A B is invertible and In B A v 0 we need to show v 0 Multiply by A to get A In B A v A v A B A v Im A B A v so A v 0 since Im A B is invertible But then v B A v 0 as desired Thus Im A B invertible implies In B A invertible The converse follows by the same argument Solution 2 From the practice final we know A B and B A have the identical nonzero eigenvalues Therefore 1 is an eigenvalue of A B if and only if it an eigenvalue of B A implying 0 is an eigenvalue of Im A B if and only if it is an eigenvalue of In B A so that Im A B is singular if and only if In B A is singular Question 2 10 points version 2 Let X be an m by n real matrix and let Y be an n by m complex matrix Show that Im X Y is invertible if and only if In Y X is invertible Answer Solution 1 Suppose Im X Y is invertible and In Y X v 0 we need to show v 0 Multiply by X to get X In Y X v X v X Y X v Im X Y X v so X v 0 since Im X Y is invertible But then v Y X v 0 as desired Thus Im X Y invertible implies In Y X invertible The converse follows by the same argument Solution 2 From the practice final we know X Y and Y X have the identical nonzero eigenvalues Therefore 1 is an eigenvalue of X Y if and only if it an eigenvalue of Y X implying 0 is an eigenvalue of Im X Y if and only if it is an eigenvalue of In Y X so that Im X Y is singular if and only if In Y X is singular 3 Question 3 10 points version 1 Let A PR L U PC be an LU decomposition of the m by n real matrix A of rank r 0 Thus PR and PC are permutation matrices L is m by r and unit lower triangular and U is r by n and upper triangular with Uii nonzero Show how to express an LU decomposition of At using simple modi cations of the parts of this LU decomposition of A Answer Write U D U where D is r by r and diagonal with Dii Uii so U is unit upper triangular Then A PR L U PC PR L D U PC so At PCt U t D Lt PRt This is an LU decomposition of At Question 3 10 points version 2 Let B PR L U PC be an LU decomposition of the m by n complex matrix B of rank r 0 Thus PR and PC are permutation matrices L is m by r and unit lower triangular and U is r by n and upper triangular with Uii nonzero Show how to express an LU decomposition of B using simple modi cations of the parts of this LU decomposition of B Answer Write U D U where D is r by r and diagonal with Dii Uii so U is unit upper triangular Then B PR L U PC PR L D U PC so B PC U D L PR This is an LU decomposition of B 4 Question 4 10 points version 1 Let T V V be a linear operator Suppose that T vi i vi for i 1 m and all i are distinct If W is an invariant subspace of T and includes the vector m i 1 ai vi where all the ai 0 then prove that W contains vi for i 1 m n …