# Register now to access 7 million high quality study materials (What's Course Hero?) Course Hero is the premier provider of high quality online educational resources. With millions of study documents, online tutors, digital flashcards and free courseware, Course Hero is helping students learn more efficiently and effectively. Whether you're interested in exploring new subjects or mastering key topics for your next exam, Course Hero has the tools you need to achieve your goals.

6 Pages

### sample1finalsol103

Course: MATH 103, Fall 2009
School: Stanford
Rating:

Word Count: 1075

#### Document Preview

103 MATH SAMPLE FINAL SOLUTIONS 1. Consider the data points (-1, 15), (0, 8), (1, 5), (2, 0). (a) Find the least-squares line for the data points above. Solution. Let f (x) = c0 + c1 x. Then c0 and c1 must satisfy c0 - c1 c0 c0 + c1 c0 + 2c1 This system is inconsistent. Since AT A = 4 2 2 6 4 2 2 6 Its reduced row echelon form is 1 0 0 1 Thus the least-squares line is f (x) = 47 24 - x 5 5 47/5 -24/5 AT b = 28 -10...

Register Now

#### Unformatted Document Excerpt

Coursehero >> California >> Stanford >> MATH 103

Course Hero has millions of student submitted documents similar to the one
below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

Course Hero has millions of student submitted documents similar to the one below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.
103 MATH SAMPLE FINAL SOLUTIONS 1. Consider the data points (-1, 15), (0, 8), (1, 5), (2, 0). (a) Find the least-squares line for the data points above. Solution. Let f (x) = c0 + c1 x. Then c0 and c1 must satisfy c0 - c1 c0 c0 + c1 c0 + 2c1 This system is inconsistent. Since AT A = 4 2 2 6 4 2 2 6 Its reduced row echelon form is 1 0 0 1 Thus the least-squares line is f (x) = 47 24 - x 5 5 47/5 -24/5 AT b = 28 -10 = 15 = 8 = 5 = 0 the augmented matrix for the normal equations is 28 -10 (b) Plot the points and the least-squares approximation. Solution. Least-Squares Line 20 15 10 5 0 -5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 (c) Find a cubic polynomial of the form f (x) = c0 + c1 x + c2 x2 + c3 x3 which passes through all four points. Solution. The coefficients must satisfy the system Ax = b where 1 -1 1 -1 15 1 0 0 0 8 A= b= 1 1 1 1 5 1 2 4 8 0 The solution is 8 -4 x= 2 -1 so f (x) = 8 - 4x + 2x2 - x3 . 2. Find an orthonormal basis for the subspace of 2 1 2 1 , , 2 3 2 3 R4 spanned by the following vectors: 0 2 . 4 6 Solution. Call these vectors v1 , v2 and v3 . Normalizing the first vector gives 1/2 1/2 v1 w1 = = 1/2 v1 1/2 Next, 1 2 -1 1 2 -1 v2 - (v2 w1 )w1 = v2 - 4w1 = - = 3 2 1 3 2 1 -1/2 -1/2 w2 = 1/2 1/2 so normalizing gives Finally 0 3 -2 -1 2 3 -2 1 v3 - (v3 w1 )w1 - (v3 w2 )w2 = v3 - 6w1 - 4w2 = - - = 4 3 2 -1 6 3 2 1 so normalizing gives -1/2 1/2 w3 = -1/2 1/2 3. Let V be any subspace of Rn . Let P1 denote the matrix for the orthogonal projection onto V and let P2 denote the matrix for the orthogonal projection onto V . (a) Show that P1 and P2 are symmetric. Hint: Use Fact 5.3.10. Solution. By Fact 5.3.10, P1 takes the form AAT for some matrix A. Thus P T = (AAT )T = (AT )T AT = AAT = P1 , so P1 is symmetric. Similarly P2 is symmetric. (b) Show that P1 P2 = P2 P1 = 0 (the zero matrix). Hint: Explain why the columns (and rows) of P1 are vectors in V . Where are the columns and rows of P2 ? Solution. The image of P1 must be V . But the image of P1 is the span of the columns of P1 . Thus each column of P1 is a vector in V , and since P1 is symmetric by part (a), the rows of P1 are also vectors in V . Likewise the rows and columns of P2 are vectors in V . Each entry in P1 P2 is the dot product of a row of P1 with a column of P2 , so each entry is the dot product of a vector in V with a vector in V . By definition of V , all such dot products are zero. Hence P1 P2 = 0, and similarly P2 P1 = 0. (c) Show that P1 + P2 = In . Hint: For any x Rn , consider w = x - P1 x. Where is w? Now apply P2 to both sides. Solution. Let x Rn . By Fact 5.1.6, the projection of x onto V is the unique vector v such that x - v is in V . Since v = P1 x this means that w = x - P1 x is in V . Applying P2 to both sides gives P2 w = P2 x - P2 P1 x = P2 x But since w is in V , P2 w = w, so w = P2 x. Thus x = P1 x + P2 x = (P1 + P2 )x for all x Rn . This means that P1 + P2 is the matrix for the identity transformation, i.e. P1 + P2 = In . 4. Let V = {x R4 | x1 + x2 + x3 + x4 = 0}. (a) Find a basis for V . Solution. 1 1 V = Span 1 1 = V 1 1 = Span 1 1 (b) Find the matrix (in standard coordinates) for the orthogonal projection onto V . Solution. Since V is spanned by the unit vector 1/2 1/2 u= 1/2 1/2 the matrix for the orthogonal projection onto V is (by Fact 5.3.10) 1 1 1 1 1 1 1 1 1 P2 = uuT = 4 1 1 1 1 1 1 1 1 (c) Find the matrix (in standard coordinates) for the orthogonal projection onto V . Hint: Use Question 3c. Solution. By Question 3c, the matrix for the orthogonal projection onto V is 3 -1 -1 -1 1 -1 3 -1 -1 P1 = I4 - P2 = -1 -1 3 -1 4 -1 -1 -1 3 5. Let 1 1 0 A= 1 0 1 0 1 1 (a) Find all eigenvalues of A and a basis for the eigenspace associated with each eigenvalue. Solution. The characteristic polynomial is fA () = ( - 1)( - 2)( + 1), so the eigenvalues are = 1, 2, -1. The eigenspaces are -1 1 1 0 E2 = Span 1 E-1 = Span -2 E1 = Span 1 1 1 (b) Diagonalize A. That is, find an invertible matrix S that S -1 AS = D. Solution. -1 1 1 S = 0 1 -2 D= 1 1 1 and a diagonal matrix D such 1 0 0 0 2 0 0 0 -1 6 0. (c) Solve the system x(t + 1) = Ax(t) with initial data x0 = 0 Solution. Since 6 -1 1 1 0 = -3 0 + 2 1 + 1 -2 0 1 1 1 the solution is 1 1 -1 0 + 2(2)t 1 + (-1)t -2 x(t) = -3 1 1 1 6. Consider the plane V = {x R3 | x1 + x2 + x3 = 0}. (a) Find a basis {v1 , v2 } for V and a basis {v3 } for V . Solution. -1 -1 1 1 , 0 1 V = Span V = Span 0 1 1 (b) Let T : R3 R3 be the linear transformation defined by reflection through V . That is, T sends each vector in R3 to its mirror image on the opposite side of V . Write down the matrix B for T with respect to the basis B = {v1 , v2 , v3 } of R3 . Solution. T (v1 ) = v1 = 1v1 + 0v2 + 0v3 T (v2 ) = v2 = 0v1 + 1v2 + 0v3 T (v3 ) = -v1 = 0v1 + 0v2 + (-1)v3 Thus 1 0 0 0 B= 0 1 0 0 -1 (c) Write down the matrix A for T with respect to the standard basis for R3 . Solution. Let -1 -1 1 0 ...

Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education.

Below is a small sample set of documents:

Stanford - CS - 103
CS103X: Discrete Structures Homework Assignment 2: SolutionsDue February 1, 2008Exercise 1 (10 Points). Prove or give a counterexample for the following: Use the Fundamental Theorem of Arithmetic to prove that for n N, n is irrational unless n i
Columbia - JM - 3058
As a post-doctoral fellow at Columbia University under Professor Michael Weinstein, the applicant will pursue further studies of linear and nonlinear dispersive equations. Such studies will include stability theory of solitons, describing the spectru
Columbia - IEOR - 3106
IEOR 3106: Introduction to Operations Research: Stochastic Models Fall 2006, Professor Whitt, Final Exam Chapters 4-7 and 10 in Ross, Tuesday, December 19, 4:10pm-7:00pm Open Book: but only the Ross textbook, three 8 11 pages of notes, and the class
Columbia - MH - 2078
IEOR E4706: Financial Eng: Discrete-Time Asset PricingColumbia University Instructor: Martin Haugh Midterm Examination October 26th 2004 Total Marks: 100 Time: 2 hours Notes that are on either side of a single letter size sheet of paper may be used
Columbia - MH - 2078
Financial Engineering I, IEOR E4706, Fall 2003Columbia University Instructor: Martin Haugh Final Examination, December 16th 2003 Total Marks: 100 Time: 3 hours Notes that are on both sides of a single sheet of A4 paper may be used during the exam. Q
Columbia - FE - 2078
Financial Engineering I, IEOR E4706, Fall 2003Columbia University Instructor: Martin Haugh Final Examination, December 16th 2003 Total Marks: 100 Time: 3 hours Notes that are on both sides of a single sheet of A4 paper may be used during the exam. Q
Stanford - EE - 392
Ellipsoid Method ellipsoid method convergence proof inequality constraints feasibility problemsProf. S. Boyd, EE392o, Stanford UniversityChallenges in cutting-plane methods can be difficult to compute appropriate next query point localiza
Stanford - MSANDE - 310
Yinyu Ye, MS&amp;E, StanfordMS&amp;E310 Lecture Note: Ellipsoid Method1The Ellipsoid (Kachiyan) MethodYinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A.http:/www.stanford.edu/ yyyeYinyu Ye, MS
SUNY Buffalo - UB - 2020
ITSM Tool Selection#3.1.1 3.1.2 3.1.3 3.1.4 3.1.5 3.1.6 3.1.7 3.1.8 3.2.1 3.2.2 3.2.3 3.2.4 3.2.5 3.2.6 3.3.1 3.3.1.1 3.3.1.2 3.3.1.3 3.3.2 3.3.2.1 3.3.2.2 3.3.3 3.3.3.1 3.3.3.2 3.3.3.3 3.3.4 3.3.4.1 3.3.4.2 3.3.4.3 3.3.4.4 3.3.4.5 3.3.4.6 3.4.1G
DISCUSSION PAPERS IN ECONOMICSWorking Paper No. 05-07 Environmental Injustice and Residential Segregation: Investigating the LinkJosh SidonUniversity of Colorado at Boulder Boulder, ColoradoOctober 24, 2005Center for Economic Analysis Departm
DISCUSSION PAPERS IN ECONOMICSWorking Paper No. 03-19Estimating Stochastic Frontier Tax Potential: Can Indonesian Local Governments Increase Tax Revenues Under Decentralization?Luky AlfirmanDepartment of Economics, University of Colorado at Boul
DISCUSSION PAPERS IN ECONOMICSWorking Paper No. 03-12U.S. Affiliates, Infrastructure and Growth: A Simultaneous Investigation of Critical MassDerek K. KellenbergDepartment of Economics, University of Colorado at Boulder Boulder, ColoradoOctobe
DISCUSSION PAPERS IN ECONOMICSWorking Paper No. 03-03 Anarchy and Demand for the State in a Trade EnvironmentAnna Rubinchik-PessachDepartment of Economics, University of Colorado at Boulder Boulder, ColoradoRoberto M. SamaniegoDepartment of Ec
DISCUSSION PAPERS IN ECONOMICSWorking Paper No. 07-08Intrametropolitan Decentralization: Overlapping Jurisdictions and Efficient Local Public Good ProvisionStephen BillingsUniversity of ColoradoNovember 15, 2007Center for Economic Analysis
DISCUSSION PAPERS IN ECONOMICSWorking Paper No. 04-05On the Efficacy of Reforms: Policy Tinkering, Institutional Change, and EntrepreneurshipMurat IyigunDepartment of Economics, University of Colorado at Boulder Boulder, ColoradoDani RodrikJo
DISCUSSION PAPERS IN ECONOMICSWorking Paper No. 08-04Multinationals and the Shutdown of State-Owned EnterprisesGrzegorz PacUniversity of ColoradoOctober 2008Center for Economic Analysis Department of EconomicsUniversity of Colorado at Bou
DISCUSSION PAPERS IN ECONOMICSWorking Paper No. 03-13Modeling Network Externalities, Network Effects, and Product Compatibility with Logit Demand(Revision of Working Paper No. 03-06) Collin StarkweatherDepartment of Economics, University of Colo
DISCUSSION PAPERS IN ECONOMICSWorking Paper No. 05-08 IPRs and Tariff Policies: East-West Joint VenturesXiaofei Vivian YangUniversity of Colorado at Boulder Boulder, ColoradoNovember 9, 2005Center for Economic Analysis Department of Economics
DISCUSSION PAPERS IN ECONOMICSWorking Paper No. 05-04Community Income Distributions in a Metropolitan AreaCharles A. M. de BartolomeUniversity of Colorado at Boulder andStephen L. RossUniversity of ConnecticutJuly 2005Center for Economic
DISCUSSION PAPERS IN ECONOMICSWorking Paper No. 04-01Building the Family Nest: A Collective Household Model with Competing Pre-marital Investments and Spousal MatchingMurat F. IyigunDepartment of Economics, University of Colorado at Boulder Boul
DISCUSSION PAPERS IN ECONOMICSWorking Paper No. 05-06An Explanation of OECD Factor Trade with Knowledge Capital and the HOV ModelShuichiro Nishioka University of Colorado at BoulderOctober, 2005 Revised: October, 2006Center for Economic Anal
DISCUSSION PAPERS IN ECONOMICSWorking Paper No. 04-04Marketing InnovationYongmin ChenDepartment of Economics, University of Colorado at Boulder Boulder, ColoradoFebruary 2004Center for Economic Analysis Department of EconomicsUniversity of
DISCUSSION PAPERS IN ECONOMICSWorking Paper No. 04-02Stated Preference Analysis of Public Goods: Are we asking the right question?Nicholas E. FloresDepartment of Economics, University of Colorado at Boulder Boulder, ColoradoAaron StrongDepart
DISCUSSION PAPERS IN ECONOMICSWorking Paper No. 07-04Does Access to Family Planning Services Improve Women's Welfare? Evidence on Dowries and IntraHousehold Bargaining in BangladeshChristina PetersUniversity of ColoradoOctober 2007Center fo