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### hw3

Course: MATH 260, Spring 2012
School: UPenn
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260, Math Spring 2012 Jerry L. Kazdan Problem Set 3 Due: In class Thursday, Feb. 2. Late papers will be accepted until 1:00 PM Friday. 1. Say you have k linear algebraic equations in n variables; in matrix form we write AX = Y . Give a proof or counterexample for each of the following. a) If n = k there is always at most one solution. b) If n &gt; k you can always solve AX = Y . c) If n &gt; k the...

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260, Math Spring 2012 Jerry L. Kazdan Problem Set 3 Due: In class Thursday, Feb. 2. Late papers will be accepted until 1:00 PM Friday. 1. Say you have k linear algebraic equations in n variables; in matrix form we write AX = Y . Give a proof or counterexample for each of the following. a) If n = k there is always at most one solution. b) If n > k you can always solve AX = Y . c) If n > k the nullspace of A has dimension greater than zero. d) If n < k then for some Y there is no solution of AX = Y . e) If n < k the only solution of AX = 0 is X = 0. 2. Let A and B be n n matrices with AB = 0. Give a proof or counterexample for each of the following. a) BA = 0 b) Either A = 0 or B = 0 (or both). c) If B is invertible then A = 0. d) There is a vector V = 0 such that BAV = 0. 3. Consider the system of equations x+y z = a x y + 2z = b. a) Find the general solution of the homogeneous equation. b) A particular solution of the inhomogeneous equations when a = 1 and b = 2 is x = 1, y = 1, z = 1. Find the most general solution of the inhomogeneous equations. c) Find some particular solution of the inhomogeneous equations when a = 1 and b = 2. d) Find some particular solution of the inhomogeneous equations when a = 3 and b = 6. [Remark: After you have done part a), it is possible immediately to write the solutions to the remaining parts.] 4. Let A = 1 1 1 1 1 2 . a) Find the general solution Z of the homogeneous equation AZ = 0. 1 b) Find some solution of AX = 1 2 c) Find the general solution of the equation in part b). d) Find some solution of AX = 1 2 e) Find some solution of AX = 3 0 f ) Find some solution of AX = 7 . [Note: ( 7 ) = ( 1 ) + 2 ( 3 )]. 2 2 0 2 and of AX = 3 6 [Remark: After you have done parts a), b) and e), it is possible immediately to write the solutions to the remaining parts.] 5. Let A : R3 R2 and B : R2 R3 , so BA : R3 R3 and AB : R2 R2 . a) Show that BA can not be invertible. b) Give an example showing that AB might be invertible. 6. Given the ve data points: P1 = (1, 1), P2 = (0, 0), P3 = (1, 0), P4 = (2, 2), P5 = (3, 0), nd the (unique!) quartic polynomial p(x) that passes through these points. [Dont bother to simplify your answer.] The next sequence of problems involve techniques for explicitely solving the ordinary dierential equation Lu := a(x)u + b(x)u + c(x)u = f (x) in the very special (but important) special case where the coecients a(x) , b(x) , and c(x) are constants with a = 0 , and the right hand side f (x) is simple. These assume you have mastered the ideas concerning the complex exponential ex+iy in http://www.math.upenn.edu/ kazdan/260S12/hw/hw0.pdf 7. Homogeneous equation example a) Let Lu := u + u 2u . Find two linearly independent solutions of the homogeneous equation of the form u(x) = erx where r is a constant, possibly complex. b) Seek (and nd) solutions u(t) of u + 2u + 5u = 0 in the form u(t) = erx , where r might be a complex number. Use this to nd two linearly independent real solutions. [See Homework Set 0]. c) Find a solution of u + 2u + 5u = 0 that satises the initial conditions u(0) = 1, u (0) = 0. 2 8. Homogeneous Let equation Lu := au + bu + cu where the coecients are real constants with a = 0. Show that L(erx ) = p(r)erx , where p(r) is a quadratic polynomial. Assume the roots of p(r) = 0 are distinct a) Find two linearly independent solutions (possibly complex) of the homogeneous equation Lu = 0. b) If the solutions you just found are complex-valued functions, use them to nd two linearly independent real solutions. 9. Inhomogeneous equation Example: Find some particular solution v (x) of v + v = x2 1. [Suggestion: Since the right hand side is a quadratic polynimial and the coecients of the dierential equation are constants, seek v (x) as a quadratic polynomial: v (x) = A + Bx + Cx2 ]. This experiment leads one to the general approach of the next problem on nding a particular solution of the inhomogeneous equation when f (x) is a polynomial. 10. Inhomogeneous equation: polynomial Let PN be the linear space of polynomials of degree at most N and L : PN PN the linear map dened by Lu := au + bu + cu , where a , b , and c are constants. Assume c = 0 (and a = 0). a) Compute L(xk ). b) Show that nullspace (=kernel) of L : PN PN is 0. [A strict proof uses induction but it is convincing enough to treat the case N = 3.] c) Show that for every polynomial q (x) PN there is one and only one solution p(x) PN of the ODE Lp = q . [A strict proof uses induction but it is convincing enough to treat the case N = 3.] 11. Inhomogeneous equation Example: Use the observation in Problem 8 to nd particular solutions of a) u 4u = 2e3x b) u 4u = cos x [Hint: cos x is the real part of eix .] c) u 4u = cos x + 2 sin x d) u 4u = ex cos x . [Not assigned but useful.] Bonus Problems [Please give these directly to Professor Kazdan] 1-B [Error in Interpolation] Let x0 < x1 < x2 be distinct real numbers and f (x) a smooth function. In class we showed there is a unique quadratic polynomial p(x) 3 with the property that p(xj ) = f (xj ) for j = 0, 1, 2. Here you nd a formula for the error:= f (x) p(x). If b is in the open interval (x0 , x2 ) with b = xj , j = 0, 1, 2, show there is a point c (depending on b ) in the interval (x0 , x2 ) so that f (b) = p(b) + f (c) (b x0 )(b x1 )(b x2 ). 3! This estimate is related to the procedure used to nd the remainder in a Taylor polynomial. [Suggestion: Dene the constant M by f (b) = p(b) + M (b x0 )(b x1 )(b x2 ), and look at g (x) := f (x) [p(x) + M (x x0 )(x x1 )(x x2 )]. Now observe that g (x) = 0 at x0 , x1 , x2 , and b (by denition of M ).] 2-B Let Lu := u + bu + cu = 0, where b and c are constants. a) If w(x) is a solution of the homogeneous equation Lw = 0 with initial conditions w(0) = 0 and w (0) = 0, show that w(x) = 0 for all x 0. b) Make the change of variable x = t and show that as a function of t w satises: dw d2 w b + cw = 0 dt dt with w(0) = 0 and w (0) = 0. This has the same structure as the original equation, only the sign of b is ipped so by applying part a), conclude that w(x) = 0 for all x . c) Use this to state and prove a uniqueness theorem for the inhomoheneous equation Lu = f (x) with u(0) = and u (0) = [Last revised: February 9, 2012] 4
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UPenn - MATH - 260
Math 260, Spring 2012Jerry L. KazdanProblem Set 4Due: Never[Exam 1 is on Thursday, Feb.9, 12:00-1:20]Unless otherwise stated use the standard Euclidean norm.1. In R2 , dene the new norm of a vector V = (x, y ) by V := 2|x| + |y | . Show thissatises
UPenn - MATH - 260
Math 260, Spring 2012Jerry L. KazdanProblem Set 5Due: In class Thursday, Feb. 16. Late papers will be accepted until 1:00 PM Friday.Unless otherwise stated use the standard Euclidean norm.1 for x &lt; 0,. Find its Fourier series (either using trig1 fo
UPenn - MATH - 260
Math 260, Spring 2012Jerry L. KazdanProblem Set 6Due: In class Thursday, Feb. 23. Late papers will be accepted until 1:00 PM Friday.Unless otherwise stated use the standard Euclidean norm.1. Let u(x, t) be the temperature at time t at a point x on a
UPenn - MATH - 260
Math 260, Spring 2012Jerry L. KazdanProblem Set 7Due: In class Thursday, March 1. Late papers will be accepted until 1:00 PM Friday.Unless otherwise stated use the standard Euclidean norm.1. Say you have a matrix A(t) = (aij (t) whose elements aij (t
UPenn - MATH - 260
Math 260, Spring 2012Jerry L. KazdanProblem Set 8Due: NeverUnless otherwise stated use the standard Euclidean norm.1. Find a 3 3 symmetric matrix A with the property thatX, AX = x2 + 4x1 x2 x1 x3 + 2x2 x3 + 5x213for all X = (x1 , x2 , x3 ) R3 .2
UPenn - MATH - 240
Problem: Score:12345678 Total:9101112_ Please do not write above this line UNIVERSITY OF PENNSYLVANIA DEPARTMENT OF MATHEMATICS MATH 240 FINAL EXAM Monday December 21, 2009 Your name (printed) _ Signature_ Professor (circle one): Patrick Clar
UPenn - MATH - 240
1Math 240Circle one:FINAL EXAMDecember 17, 2010Professor ShatzProfessor WylieProfessor ZillerName:Penn Id#:Signature:TA:Recitation Day and Time:Please show your work clearly. A correct answer with no work is worth 0 points. Circle your answer
UPenn - MATH - 240
NAME: PROFESSOR: (A) Donagi ; (B) Ghrist ; (C) Krieger240 SPRING 2009: CalculusFINAL EXAM INSTRUCTIONS:1. WRITE YOUR NAME at the top and indicate your professors name. 2. As you solve problems on the exam, FILL IN COMPLETELY the letter(s) of your solut
UPenn - MATH - 240
1Math 240Circle one:FINAL EXAMMay 4, 2010Professor ZillerProfessor ZywinaName:Penn Id#:Signature:TA:Recitation Day and Time:You need to show your work, even for multiple choice problems. A correct answer with no workwill get 0 points. The onl
UPenn - MATH - 240
Math 240 Final Exam, spring 2011Name (printed):TA:Recitation Time:This examination consists of ten (10 problems). Please turn o all electronicdevices. You may use both sides of a 8.5 11 sheet of paper for notes whileyou take this exam. No calculator
UPenn - MATH - 240
Makeup Final Exam, Math 240: Calculus IIISeptember, 2011No books, papers or may be used, other than a hand-writtennote sheet at most 8.5 11 in size. All electronic devicesmust be switched o during the exam.This examination consists of nine (9) long a
UPenn - MATH - 425
Math 425March 3, 2011Exam 1Jerry L. Kazdan12:00 1:20Directions This exam has three parts, Part A, short answer, has 1 problem (12 points). Part Bhas 5 shorter problems (7 points each, so 35 points). Part C has 3 traditional problems (15 pointseach
UPenn - MATH - 425
Math 425March 3, 2011Exam 1Jerry L. Kazdan12:00 1:20Directions This exam has three parts, Part A, short answer, has 1 problem (12 points). Part Bhas 5 shorter problems (7 points each, so 35 points). Part C has 3 traditional problems (15 pointseach
UPenn - MATH - 425
SignaturePrinted NameMy signature above certies that I have complied with the University of PennsylvaniasCode of Academic Integrity in completing this examination.Exam 2Math 425April 26, 2011Jerry L. Kazdan12:00 1:20Directions This exam has three
UPenn - MATH - 425
SignaturePrinted NameMy signature above certies that I have complied with the University of PennsylvaniasCode of Academic Integrity in completing this examination.Exam 2Math 425April 26, 2011Jerry L. Kazdan12:00 1:20Directions This exam has three
UPenn - MATH - 425
AMSIJan. 14 Feb. 8, 2008Partial Differential EquationsJerry L. KazdanCopyright c 2008 by Jerry L. Kazdan[Last revised: March 28, 2011]ivCONTENTS7. Dirichlets principle and existence of a solutionChapter 6. The RestContentsChapter 1. Introductio
UPenn - MATH - 425
Math 425Notes and exercises on Black-ScholesDr. DeTurckApril 2010On Thursday we talked in class about how to derive the Black-Scholes dierentialequation, which is used in mathematical nance to assign a value to a nancial derivative. The latter is usu
UPenn - MATH - 425
Math 425, Spring 2011Jerry L. Kazdanwhere the coefcient matrix B := (bk ) isPDE: Linear Change of Variablenbk =Let x := (x1 , x2 , . . . , xn ) be a point inential operatorai, j ski s j .i, j=1Rnand consider the second order linear partial diff
UPenn - MATH - 425
Math 425, Spring 2011Jerry L. KazdanPDE: Linear Change of VariableLet x := (x1, x2, . . . , xn) be a point in Rn and considerthe second order linear partial differential operatorn2 uLu := ai j,xix ji, j=1(1)where the coefcient matrix A := (ai
UPenn - MATH - 425
Math 425 Midterm 1Dr. DeTurck March 2, 2010There are four problems on this test. You may use your book and your notes during this exam. Do as much of it as you can during the class period, and turn your work in at the end. But take the sheet with the pr
UPenn - MATH - 425
Math 425 Midterm 1Dr. DeTurck February 8, 20071. Solve ux + yuy + u = 0, u(0, y ) = y . In what domain in the plane is your solution valid? 2. Let u(x, t) be the temperature in a rod of length L that satises the partial dierential equation: ut = kuxx fo
UPenn - MATH - 425
Fourier Series of f (x) = xTo write:eikx.x = ck2kThe Fourier coefcients are112i ck = xeikx dx = x[cos kx i sin kx] dx = x sin kx dx2 2 2 0Butx cos kx 1 cos kx sin kx dx =+cos kx dx == (1)k .kk0kk00Thus2i(1)kck = (1)k =
UPenn - MATH - 425
Math 425, Spring 2011Jerry L. KazdanProblem Set 0 [Rust Remover]D UE : Never.1. Let u(t ) be the amount of a radioactive element at time t and say initially, u(0) = A . The rateof decay is proportional to the amount present, sodu= cu(t ),dtwhere
UPenn - MATH - 425
Math 425/525, Spring 2011Jerry L. KazdanProblem Set 1D UE : In class Thursday, Jan. 27. Late papers will be accepted until 1:00 PM Friday.1. Find Greens function g(x, s) to get a formula u(x) =of u (x) = f (x) .Rx0g(x, s) f (s) ds for a particular
UPenn - MATH - 425
Math 425/525, Spring 2011Jerry L. KazdanProblem Set 2D UE : In class Thursday, Feb. 3 Late papers will be accepted until 1:00 PM Friday.1. Find the solution U (t ) := (u1 (t ), u2 (t ) ofu1 =u1u2 =u1 u2with the initial conditions U (0) = (u1 (0), u
UPenn - MATH - 425
Math 425Assignment 3Dr. DeTurckDue Tuesday, February 9, 2010Reading: Textbook, Chapter 1, skim through Chapter 2.1. Take a moment to review the divergence theorem from vector calculus, then work thefollowing problem: Suppose V (x, y, z ) is a vector
UPenn - MATH - 425
Math 425/525, Spring 2011Jerry L. KazdanProblem Set 3D UE : In class Thursday, Feb. 10 Late papers will be accepted until 1:00 PM Friday.1. Solve ux + uy + u = ex+2y with u(x, 0) = 0.2. Find the general solution of uxy = x2 y for the function u(x, y)
UPenn - MATH - 425
Math 425/525, Spring 2011Jerry L. KazdanProblem Set 4D UE : In class Thursday, Feb. 17 Late papers will be accepted until 1:00 PM Friday.1. Solve the wave equation utt = c2 uxx for the semi-innite string x 0 with the initial andboundary conditionsu(
UPenn - MATH - 425
Math 425, Spring 2011Jerry L. KazdanProblem Set 5D UE : Thurs. Feb. 24 Late papers will be accepted until 1:00 PM Friday.1. In R4 the vectorsU1 := (1, 1, 1, 1),U2 := (1, 1, 1, 1),U3 := (2, 2, 2, 2),U4 := (1, 1, 1, 1)are orthogonal, as you can eas
UPenn - MATH - 425
Math 425, Spring 2011Jerry L. KazdanProblem Set 6D UE : Thursday March 17 [Late papers will be accepted until 1:00 PM Friday].1. This problem concerns orthogonal projections into a subspace of a larger space.a) Let U = (1, 1, 0, 1) and V = (1, 2, 1,
UPenn - MATH - 425
Math 425, Spring 2011Jerry L. KazdanProblem Set 7D UE : Thursday March 24 [Late papers will be accepted until 1:00 PM Friday].1. Suppose u is a twice differentiable function on R which satises the ordinary differentialequationu + b(x)u c(x)u = 0,wh
UPenn - MATH - 425
Math 425, Spring 2011Jerry L. KazdanProblem Set 8D UE : Thursday March 31 [Late papers will be accepted until 1:00 PM Friday].1. a) Let A be an n n invertible real symmetric matrix, b Rn and c R . For x Rn considerthe quadratic polynomialQ(x) = x, A
UPenn - MATH - 425
Math 425, Spring 2011Jerry L. KazdanProblem Set 9D UE : Thursday April 7 [Late papers will be accepted until 1:00 PM Friday].1. a) In a bounded region Rn , let u(x, t ) satisfy the modied heat equationut = u + cu,where c is a constant,(1)as well a
UPenn - MATH - 425
Math 425, Spring 2011Jerry L. KazdanProblem Set 10D UE : Thursday April 14 [Late papers will be accepted until 1:00 PM Friday].1. This problem is to help with a computation in class today (Thursday) nding a formula for aparticular solution of the inh
UPenn - MATH - 425
INTERMEDIATE CALCULUSANDLINEAR ALGEBRAPart IJ. KAZDANHarvard UniversityLecture NotesiiPrefaceThese notes will contain most of the material covered in class, and be distributed beforeeach lecture (hopefully). Since the course is an experimental o
UPenn - MATH - 425
UPenn - MATH - 425
UPenn - MATH - 425
First-order ordinary differential equations Before we get involved trying to understand partial differential equations (PDEs), we'll review the highlights of the theory of ordinary differential equations (ODEs). We'll do this in such a way that we can beg
UPenn - MATH - 425
Linear ODEsSecond order linear equationsMany traditional problems involving ordinary equations arise as second order linear equationsau + bu + cu = f ,more briey asLu = f .The problem is, given f , nd u ; often we will want to nd u that satises some
UPenn - MATH - 425
Math 425 Midterm 1Dr. DeTurck February 8, 20071. Solve ux + yuy + u = 0, u(0, y ) = y . In what domain in the plane is your solution valid? As usual, we construct the graph of the solution by propagating the initial data o the line in the xy -plane wher
UPenn - MATH - 425
Math 425/525, Spring 2011Jerry L. KazdanPeriodic Solutions of ODEsIn class we discussed some aspects of periodic solutions of ordinary differential equations. Fromthe questions I received, my presentation was not so clear. Here Ill give a detailed for
UPenn - MATH - 425
Math 425Practice Midterm 1Dr. DeTurckFebruary 20101. Suppose f is a function of one variable that has a continuous second derivative. Show thatfor any constants a and b, the functionu(x, y ) = f (ax + by )is a solution of the PDEuxx uyy u2 = 0.xy
UPenn - MATH - 425
Math 425Hints and Solutions to Practice Midterm 1Dr. DeTurckFebruary 20101. Suppose f is a function of one variable that has a continuous second derivative. Show thatfor any constants a and b, the functionu(x, y ) = f (ax + by )is a solution of the
UPenn - MATH - 425
Math 425, Spring 2011Jerry L. KazdanThe Wave Equation in R2 and R3To Solveutt = c2u,x Rnwithu(x, 0) = (x),ut (x, 0) = (x)In R3 [Poisson: Kirchoffs formula]1u(x0, t0) =4c2t01(x) dS +t0 4c2t0SZZZZ(x) dS ,Swhere S is the sphere centered
UPenn - MATH - 508
Exam 1Math 508October 12, 2006Jerry L. Kazdan12:00 1:20Directions This exam has three parts, Part A has 4 problems asking for Examples (20 points, 5points each), Part B asks you to describe some sets (20 points), Part C has 4 traditional problems(6
UPenn - MATH - 508
Exam 1Math 508October 12, 2006Jerry L. Kazdan12:00 1:20Directions This exam has three parts, Part A has 4 problems asking for Examples (20 points, 5points each), Part B asks you to describe some sets (20 points), Part C has 4 traditional problems(6
UPenn - MATH - 508
SignaturePrinted NameExam 2Math 508December 8, 2006Jerry L. Kazdan12:00 1:20Directions This exam has two parts, Part A has 3 shorter problems (8 points each, so 24 points),Part B has 5 traditional problems (15 points each, so 75 points).Closed bo
UPenn - MATH - 508
Exam 2Math 508December 8, 2006Jerry L. Kazdan12:00 1:20Directions This exam has two parts, Part A has 3 shorter problems (8 points each, so 24 points),Part B has 5 traditional problems (15 points each, so 75 points).Closed book, no calculators but
UPenn - MATH - 508
Exam 1Math 508October 16, 2008Jerry L. Kazdan10:30 11:50Directions This exam has three parts, Part A has 4 problems asking for Examples (20 points, 5points each), Part B asks you to describe some sets (20 points), Part C has 4 traditional problems(
UPenn - MATH - 508
Exam 2Math 508December 4, 2008Jerry L. Kazdan10:30 11:50Directions This exam has two parts, Part A has 10 True-False problems (30 points, 3 pointseach). Part B has 5 traditional problems (70 points, 14 points each).Closed book, no calculators or co
UPenn - MATH - 508
Exam 1Math 508October 14, 2010Jerry L. Kazdan9:00 10:20Directions This exam has three parts, Part A asks for 4 examples (20 points, 5 points each).Part B has 4 shorter problems (36 points, 9 points each. Part C has 3 traditional problems (45points,
UPenn - MATH - 508
Exam 2Math 508December 9, 2010Jerry L. Kazdan9:00 10:20Directions This exam has three parts, Part A asks for 3 examples (5 points each, so 15 points).Part B has 4 shorter problems (8 points each so 32 points). Part C has 4 traditional problems (15p
UPenn - MATH - 508
Math 508December 9, 2010Exam 2Jerry L. Kazdan9:00 10:20Directions This exam has three parts, Part A asks for 3 examples (5 points each, so 15 points).Part B has 4 shorter problems (8 points each so 32 points). Part C has 4 traditional problems (15p
UPenn - MATH - 508
Advanced Analysis: OutlineMath 508, Fall 2010Jerry L. KazdanThis outline of the course is to help you step back and get a larger view of what we havedone. Since this is only an outline, I will often not explicitly state the precise assumptionsneeded
UPenn - MATH - 508
Analysis ProblemsPenn MathJerry L. KazdanIn the following, when we say a function is smooth, we mean that all of its derivatives existand are continuous.These problems have been crudely sorted by topic but this should not be taken seriouslysince man
UPenn - MATH - 508
Some Classical InequalitiesFor all of these inequalities there are many methods. We give a sampling. 1.ARITHMETIC - GEOMETRIC MEAN and decide when equality holds. INEQUALITYIf cfw_ b j &gt; 0, prove the following (1)(b1 b2 bn )1/n b1 + b 2 + + b n . n
UPenn - MATH - 508
Basic DenitionsLet S Rn . and p Rn . S is bounded if it is contained in someball in Rn . S is a neighborhood of p if S containssome open ball around P . A point p is a limit point of S if everyneighborhood of p contains a point q S , where q = p .
UPenn - MATH - 508
Calculus ProblemsMath 504 505Jerry L. Kazdan1. Sketch the points (x , y) in the plane R2 that satisfy |y x| 2.2. A certain function f (x) has the property thatf (t ) dt = ex cos x + C . Find both f0and the constant C .cos x3. Compute limx0 cos 2
UPenn - MATH - 508
Math 508Jerry L. KazdanCompleteness of1Let 1 be the vector space of innite sequences of real numbers X = (x1 , x2 , . . .) with nite normX := =1 |x j | . Here we show this space is complete. The proof is a bit fussy.j(n)(n)S TEP 1: F IND A CANDID
UPenn - MATH - 508
Contracting Maps and an ApplicationMath 508, Fall 2010Jerry L. KazdanOne often effective way to show that an equation g(x) = b has a solution is to reduce theproblem to nd a xed point x of a contracting map T , so T x = x . For instance, assumeV is a
UPenn - MATH - 508
ConvolutionLet f (x) and g(x) be continuous real-valued functions for x R and assume that f or g is zerooutside some bounded set (this assumption can be relaxed a bit). Dene the convolution( f g)(x) :=Zf (x y)g(y) dy(1)Since f or g is zero outside