Elementary Linear Algebra, 6th Edition - Kau
INDEX OF APPLICATIONSBIOLOGY AND LIFE SCIENCESCalories burned, 117Populationof deer, 43of rabbits, 459Population growth, 458– 461, 472, 476, 477Reproduction rates of deer, 115Spread of a virus, 112BUSINESS AND ECONOMICSAverage monthly cable television rates, 119Basic cable and satellite television, 173Cable television service, 99, 101Consumer preference model, 99, 101, 174Consumer Price Index, 119Demandfor a certain grade of gasoline, 115for a rechargeable power drill, 115Economic system, 107Industries, 114, 119Market research, 112Net profitMicrosoft, 38Polo Ralph Lauren, 335Number of storesTarget Corporation, 354Production levelsguitars, 59vehicles, 59Profit from crops, 59Retail sales of running shoes, 354RevenueeBay, Inc., 354Google, Inc., 354Sales, 43Advanced Auto Parts, 334Auto Zone, 334Circuit City Stores, 355Dell, Inc., 335Gateway, Inc., 334Wal-Mart, 39Subscribers of a cellular communications company, 170Total cost of manufacturing, 59COMPUTERS AND COMPUTER SCIENCEComputer graphics, 410 – 413, 415, 418Computer operator, 142ELECTRICAL ENGINEERINGCurrent flow in networks, 33, 36, 37, 40, 44Kirchhoff’s Laws, 35, 36MATHEMATICSArea of a triangle, 164, 169, 173Collinear points, 165, 169Conic sections and rotation, 265–270, 271–272, 275Coplanar points, 167, 170Equationof a line, 165–166, 170, 174of a plane, 167–168, 170, 174Fourier approximations, 346–350, 351–352, 355Linear differential equations in calculus, 262–265,270 –271, 274 –275Quadratic forms, 463– 471, 473, 476Systems of linear differential equations, 461– 463,472– 473, 476Volume of a tetrahedron, 166, 170MISCELLANEOUSCarbon dioxide emissions, 334Cellular phone subscribers, 120College textbooks, 170Doctorate degrees, 334Fertilizer, 119Final grades, 118Flowof traffic, 39, 40of water, 39Gasoline, 117Milk, 117Motor vehicle registrations, 115Networkof pipes, 39of streets, 39, 40
Population, 118, 472, 476, 480of consumers, 112of smokers and nonsmokers, 112of the United States, 38Projected population of the United States, 173Regional populations, 60Television viewing, 112Voting population, 60World population, 330NUMERICAL LINEAR ALGEBRAAdjoint of a matrix, 158–160, 168–169, 173Cramer’s Rule, 161–163, 169–170, 173Cross product of two vectors in space, 336–341, 350 –351,355Cryptography, 102, 113–114, 118–119Geometry of linear transformations in the plane, 407– 410,413–414, 418Idempotent matrix, 98Leontief input-output models, 105, 114, 119LU-factorization, 93–98, 116–117QR-factorization, 356–357Stochastic matrices, 98, 118PHYSICAL SCIENCESAstronomy, 332Average monthly temperature, 43Periods of planets, 31World energy consumption, 354SOCIAL AND BEHAVIORAL SCIENCESSportsaverage salaries of Major League Baseball players, 120average salary for a National Football League player,354basketball, 43Fiesta Bowl Championship Series, 41Super Bowl I, 43Super Bowl XLI, 41Test scores, 120 –121STATISTICSLeast squares approximations, 341–346, 351, 355Least squares regression analysis, 108, 114 –115, 119–120
Elementary Linear AlgebraSIXTH EDITIONRON LARSONThe Pennsylvania State UniversityThe Behrend CollegeDAVI D C. FALVOThe Pennsylvania State UniversityThe Behrend CollegeHOUGHTON MIFFLIN HARCOURT PUBLISHING COMPANYBostonNew York
Publisher: Richard StrattonSenior Sponsoring Editor: Cathy CantinSenior Marketing Manager: Jennifer JonesDiscipline Product Manager: Gretchen Rice KingAssociate Editor: Janine TangneyAssociate Editor: Jeannine LawlessSenior Project Editor: Kerry FalveyProgram Manager: Touraj ZadehSenior Media Producer: Douglas WinickiSenior Content Manager: Maren KunertArt and Design Manager: Jill HaberCover Design Manager: Anne S. KatzeffSenior Photo Editor: Jennifer Meyer DareSenior Composition Buyer: Chuck DuttonNew Title Project Manager: Susan PeltierManager of New Title Project Management: Pat O’NeillEditorial Assistant: Amy HainesMarketing Assistant: Michael MooreEditorial Assistant: Laura CollinsCover image: Carl Reader/age fotostockCopyright 2009 by Houghton Mifflin Harcourt Publishing Company.All rights reserved.No part of this work may be reproduced or transmitted in any form or by any means,electronic or mechanical, including photocopying and recording, or by any informationstorage or retrieval system without the prior written permission of Houghton MifflinHarcourt Publishing Company unless such copying is expressly permitted by federalcopyright law. Address inquiries to College Permissions, Houghton Mifflin HarcourtPublishing Company, 222 Berkeley Street, Boston, MA 02116-3764.Printed in the U.S.A.Library of Congress Control Number: 2007940572Instructor’s examination copyISBN-13: 978-0-547-00481-5ISBN-10: 0-547-00481-8For orders, use student text ISBNsISBN-13: 978-0-618-78376-2ISBN-10: 0-618-78376-8123456789-DOC-12 11 10 09 08
ContentsCHAPTER 11.11.21.3CHAPTER 22.12.22.32.42.5A WORD FROM THE AUTHORSviiWHAT IS LINEAR ALGEBRA?xvSYSTEMS OF LINEAR EQUATIONS1Introduction to Systems of Linear EquationsGaussian Elimination and Gauss-Jordan EliminationApplications of Systems of Linear Equations11429Review ExercisesProject 1 Graphing Linear EquationsProject 2 Underdetermined and Overdetermined Systems of Equations414445MATRICES46Operations with MatricesProperties of Matrix OperationsThe Inverse of a MatrixElementary MatricesApplications of Matrix Operations4661738798Review ExercisesProject 1 Exploring Matrix MultiplicationProject 2 Nilpotent Matrices115120121iii
ivContentsCHAPTER 33.13.23.33.43.5CHAPTER 44.14.24.34.44.54.64.74.8CHAPTER 55.15.25.35.45.5DETERMINANTS122The Determinant of a MatrixEvaluation of a Determinant Using Elementary OperationsProperties of DeterminantsIntroduction to EigenvaluesApplications of Determinants122132142152158Review ExercisesProject 1 Eigenvalues and Stochastic MatricesProject 2 The Cayley-Hamilton TheoremCumulative Test for Chapters 1–3171174175177VECTOR SPACES179nVectors in RVector SpacesSubspaces of Vector SpacesSpanning Sets and Linear IndependenceBasis and DimensionRank of a Matrix and Systems of Linear EquationsCoordinates and Change of BasisApplications of Vector Spaces179191198207221232249262Review ExercisesProject 1 Solutions of Linear SystemsProject 2 Direct Sum272275276INNER PRODUCT SPACES277nLength and Dot Product in RInner Product SpacesOrthonormal Bases: Gram-Schmidt ProcessMathematical Models and Least Squares AnalysisApplications of Inner Product Spaces277292306320336Review ExercisesProject 1 The QR-FactorizationProject 2 Orthogonal Matrices and Change of BasisCumulative Test for Chapters 4 and 5352356357359
ContentsCHAPTER 66.16.26.36.46.5CHAPTER 77.17.27.37.4CHAPTER 88.18.28.38.48.5vLINEAR TRANSFORMATIONS361Introduction to Linear TransformationsThe Kernel and Range of a Linear TransformationMatrices for Linear TransformationsTransition Matrices and SimilarityApplications of Linear Transformations361374387399407Review ExercisesProject 1 Reflections in the Plane (I)Project 2 Reflections in the Plane (II)416419420EIGENVALUES AND EIGENVECTORS421Eigenvalues and EigenvectorsDiagonalizationSymmetric Matrices and Orthogonal DiagonalizationApplications of Eigenvalues and Eigenvectors421435446458Review ExercisesProject 1 Population Growth and Dynamical Systems (I)Project 2 The Fibonacci SequenceCumulative Test for Chapters 6 and 7474477478479COMPLEX VECTOR SPACES (online)*Complex NumbersConjugates and Division of Complex NumbersPolar Form and DeMoivre's TheoremComplex Vector Spaces and Inner ProductsUnitary and Hermitian MatricesReview ExercisesProject Population Growth and Dynamical Systems (II)
viContentsCHAPTER 99.19.29.39.49.5LINEAR PROGRAMMING (online)*Systems of Linear InequalitiesLinear Programming Involving Two VariablesThe Simplex Method: MaximizationThe Simplex Method: MinimizationThe Simplex Method: Mixed ConstraintsReview ExercisesProject Cholesterol LevelsCHAPTER 1010.110.210.310.4NUMERICAL METHODS (online)*Gaussian Elimination with Partial PivotingIterative Methods for Solving Linear SystemsPower Method for Approximating EigenvaluesApplications of Numerical MethodsReview ExercisesProject Population GrowthAPPENDIXMATHEMATICAL INDUCTION AND OTHERFORMS OF PROOFSA1ONLINE TECHNOLOGY GUIDE (online)*ANSWER KEYINDEX*Available online at college.hmco.com/pic/larsonELA6e.A9A59
A Word from the AuthorsWelcome! We have designed Elementary Linear Algebra, Sixth Edition, for theintroductory linear algebra course.Students embarking on a linear algebra course should have a thorough knowledge ofalgebra, and familiarity with analytic geometry and trigonometry. We do not assume thatcalculus is a prerequisite for this course, but we do include examples and exercises requiring calculus in the text. These exercises are clearly labeled and can be omitted if desired.Many students will encounter mathematical formalism for the first time in this course.As a result, our primary goal is to present the major concepts of linear algebra clearly andconcisely. To this end, we have carefully selected the examples and exercises to balancetheory with applications and geometrical intuition.The order and coverage of topics were chosen for maximum efficiency, effectiveness,and balance. For example, in Chapter 4 we present the main ideas of vector spaces andbases, beginning with a brief look leading into the vector space concept as a natural extension of these familiar examples. This material is often the most difficult for students, butour approach to linear independence, span, basis, and dimension is carefully explained andillustrated by examples. The eigenvalue problem is developed in detail in Chapter 7, but welay an intuitive foundation for students earlier in Section 1.2, Section 3.1, and Chapter 4.Additional online Chapters 8, 9, and 10 cover complex vector spaces, linear programming, and numerical methods. They can be found on the student website for this text atcollege.hmco.com/pic/larsonELA6e.Please read on to learn more about the features of the Sixth Edition.We hope you enjoy this new edition of Elementary Linear Algebra.vii
viiiA Word from the AuthorsAcknowledgmentsWe would like to thank the many people who have helped us during various stages of theproject. In particular, we appreciate the efforts of the following colleagues who made manyhelpful suggestions along the way:Elwyn Davis, Pittsburg State University, VAGary Hull, Frederick Community College, MDDwayne Jennings, Union University, TNKarl Reitz, Chapman University, CACindia Stewart, Shenandoah University, VARichard Vaughn, Paradise Valley Community College, AZCharles Waters, Minnesota State University–Mankato, MNDonna Weglarz, Westwood College–DuPage, ILJohn Woods, Southwestern Oklahoma State University, OKWe would like to thank Bruce H. Edwards, The University of Florida, for hiscontributions to previous editions of Elementary Linear Algebra.We would also like to thank Helen Medley for her careful accuracy checking of thetextbook.On a personal level, we are grateful to our wives, Deanna Gilbert Larson and SusanFalvo, for their love, patience, and support. Also, special thanks go to R. Scott O’Neil.Ron LarsonDavid C. Falvo
Proven Pedagogy Integrated Technology Real-World ApplicationsTheorems and ProofsTHEOREM 2.9The Inverseof a ProductTheorems are presented in clear and mathematicallyprecise language.Key theorems are also available via PowerPoint Presentation on the instructor website. They canbe displayed in class using a computer monitor orprojector, or printed out for use as class handouts.If A and B are invertible matrices of size n, then AB is invertible and AB 1 B 1A 1.Students will gain experience solving proofspresented in several different ways: Some proofs are presented in outline form, omittingthe need for burdensome calculations. Specialized exercises labeled Guided Proofs leadstudents through the initial steps of constructingproofs and then utilizing the results. The proofs of several theorems are left as exercises,to give students additional practice.PROOF EB E B .A full listing of the applications can be found in theIndex of Applications inside the front cover. This can be generalized to conclude that Ek . .Ei is an elementary matrix. Now consider theTheorem 2.14, it can be written as the productand you can write . E2E1B Ek . . . E2 E1 B , wherematrix AB. If A is nonsingular, then, byof elementary matrices A Ek . . . E2E1. . E E3.9:AB ProveEk .Theorem56. Guided Proof2 1B If A is a square matrix, thenTdet A det A . E . . . E E B E . . . E E B Ak21k2 1Getting Started: To prove that the determinants of A and ATare equal, you need to show that their cofactor expansions areequal. Because the cofactors are determinants of smallermatrices, you need to use mathematical induction. B .(i) Initial step for induction: If A is of order 1, then A a11 AT, so det A det AT a11.(ii) Assume the inductive hypothesis holds for all matricesof order n 1. Let A be a square matrix of order n.Write an expression for the determinant of A byexpanding by the first row.(iii) Write an expression for the determinant of AT byexpanding by the first column.(iv) Compare the expansions in (i) and (ii). The entries ofthe first row of A are the same as the entries of the firstcolumn of AT. Compare cofactors (these are the determinants of smaller matrices that are transposes ofone another) and use the inductive hypothesis toconclude that they are equal as well.Real World ApplicationsREVISED! Each chapter ends with a section onreal-life applications of linear algebra concepts,covering interesting topics such as: Computer graphics Cryptography Population growth and more!To begin, observe that if E is an elementary matrix, then, by Theorem 3.3, the next few statements are true. If E is obtained from I by interchanging two rows, then E 1. IfE is obtained by multiplying a row of I by a nonzero constant c, then E c. If E isobtained by adding a multiple of one row of I to another row of I, then E 1. Additionally,by Theorem 2.12, if E results from performing an elementary row operation on I and thesame elementary row operation is performed on B, then the matrix EB results. It follows thatEXAMPLE 4Forming Uncoded Row MatricesWrite the uncoded row matrices of size 1 3 for the message MEET ME MONDAY.SOLUTIONPartitioning the message (including blank spaces, but ignoring punctuation) into groups ofthree produces the following uncoded row matrices.[13 5M E5] [20 0 13] [5 0E T M E13] [15 14 4] [1M O N D A25 0]YNote that a blank space is used to fill out the last uncoded row matrix.INDEX OF APPLICATIONSBIOLOGY AND LIFE SCIENCESCalories burned, 117Populationof deer, 43of rabbits, 459Population growth, 458–461, 472, 476, 477Reproduction rates of deer, 115Sd fi112COMPUTERS AND COMPUTER SCIENCEComputer graphics, 410–413, 415, 418Computer operator, 142ELECTRICAL ENGINEERINGCurrent flow in networks, 33, 36, 37, 40, 44Kirchhoff’s Laws, 35, 36ix
Proven Pedagogy Integrated Technology Real-World ApplicationsConceptual UnderstandingCHAPTER OBJECTIVES Find the determinants of a 2 ⴛ 2 matrix and a triangular matrix. Find the minors and cofactors of a matrix and use expansion by cofactors to find thedeterminant of a matrix.NEW! Chapter Objectives are now listed on eachchapter opener page. These objectives highlight the keyconcepts covered in the chapter, to serve as a guide tostudent learning. Use elementary row or column operations to evaluate the determinant of a matrix. Recognize conditions that yield zero determinants. Find the determinant of an elementary matrix. Use the determinant and properties of the determinant to decide whether a matrix is singularor nonsingular, and recognize equivalent conditions for a nonsingular matrix. Verify and find an eigenvalue and an eigenvector of a matrix.The Discovery features are designedto help students develop an intuitiveunderstanding of mathematicalconcepts and relationships.True or False? In Exercises 62–65, determine whether each statement is true or false. If a statement is true, give a reason or cite anappropriate statement from the text. If a statement is false, providean example that shows the statement is not true in all cases or cite anappropriate statement from the text.62. (a) The nullspace of A is also called the solution space of A.(b) The nullspace of A is the solution space of the homogeneoussystem Ax 0.63. (a) If an m n matrix A is row-equivalent to an m n matrixB, then the row space of A is equivalent to the row spaceof B.True or False? exercises test students’knowledge of core concepts. Students areasked to give examples or justifications tosupport their conclusions.(b) If A is an m n matrix of rank r, then the dimension of thesolution space of Ax 0 is m r.DiscoveryLet 6A 01421 13 .2Use a graphing utility orcomputer software program tofind A 1. Compare det( A 1)with det( A). Make a conjectureabout the determinant of theinverse of a matrix.Graphics and Geometric EmphasisVisualization skills are necessary for the understanding of mathematical concepts andtheory. The Sixth Edition includes the following resources to help develop these skills: Graphs accompany examples, particularly when representing vector spaces andinner product spaces. Computer-generated illustrations offer geometric interpretations of problems.z42(6, 2, 4)u24x6(1, 2, 0)v2aprojvuy(2, 4, 0)zoTraceyxzEllipsoidRFigure 5.13EllipseEllipseEllipseyz-tracey2z2x2 1a2 b2 c2Planexz-traceParallel to xy-planeParallel to xz-planeParallel to yz-planeThe surface is a sphere if a b c 0.yxxy-tracex
Proven Pedagogy Integrated TechnologyReal-World ApplicationsProblem Solving and Review53. u 0, 1,54. u 1,2 , v 1,3, 2 , v 2, 1 2, 1, 1 283. u 3, 3 , v 2, 4 2 84. u 1, 1 , v 0, 1 55. u 0, 2, 2, 1, 1, 2 , v 2, 0, 1, 1, 2, 2 85. u 0, 1, 0 , v 1, 2, 0 56. u 1, 2, 3, 2, 1, 3 , v 1, 0, 2, 1, 2, 3 86. u 0, 1, 6 , v 1, 2, 1 57. u 1, 1, 2, 1, 1, 1, 2, 1 ,v 1, 0, 1, 2, 2, 1, 1, 2 87. u 2, 5, 1, 0 , v 14, 54, 0, 1 313 1188. u 4, 2, 1, 2 , v 2, 4, 2, 4 58. u 3, 1, 2, 1, 0, 1, 2, 1 ,v 1, 2, 0, 1, 2, 2, 1, 0 In Exercises 59–62, verify the Cauchy-Schwarz Inequality for thegiven vectors.59. u 3, 4 , v 2, 3 In Exercises 89–92, use a graphing utility or computer softwareprogram with vector capabilities to determine whether u and v areorthogonal, parallel, or neither.13589. u 2, 2, 1, 3 , v 2, 1, 2, 0 21 43321990. u 2 , 2 , 12, 2 , v 0, 6, 2 , 2 60. u 1, 0 , v 1, 1 3 3933 991. u 4, 2, 2, 6 , v 8, 4, 8, 3 61. u 1, 1, 2 , v 1, 3, 2 4 832164292. u 3, 3, 4, 3 , v 3 , 2, 3, 3 62. u 1, 1, 0 , v 0, 1, 1 In Exercises 63– 72, find the angle between the vectors.63. u 3, 1 , v 2, 4 Writing In Exercises 93 and 94, determine if the vectors areorthogonal, parallel, or neither. Then explain your reasoning.64. u 2, 1 , v 2, 0 93. u cos , sin , 1 , v sin , cos , 0 3 3 65. u cos , sin , v cos , sin664494. u sin , cos , 1 , v sin , cos , 0 Each chapter includes two ChapterProjects, which offer the opportunity forgroup activities or more extensive homeworkassignments.Chapter Projects are focused ontheoretical concepts or applications, andmany encourage the use of technology.CHAPTER 3REVISED! Comprehensive section and chapter exercisesets give students practice in problem-solving techniquesand test their understanding of mathematical concepts. Awide variety of exercise types are represented, including: Writing exercises Guided Proof exercises Technology exercises, indicated throughout the textwith. Applications exercises Exercises utilizing electronic data sets, indicatedbyand found on the student website atcollege.hmco.com/pic/larsonELA6eProjects1 Eigenvalues and Stochastic MatricesIn Section 2.5, you studied a consumer preference model for competing cabletelevision companies. The matrix representing the transition probabilities was 0.70P 0.200.100.150.800.05When provided with the initial state matrix X, you observed that the number ofsubscribers after 1 year is the product PX. 15,000X 20,00065,000Cumulative Tests follow chapters 3, 5,and 7, and help students synthesize theknowledge they have accumulatedthroughout the text, as well as prepare forexams and future mathematics courses. 0.150.15 .0.70 0.70PX 0.200.100.150.800.050.150.150.70 15,00023,25020,000 28,75065,00048,000CHAPTERS 4 & 5 Cumulative TestTake this test as you would take a test in class. After you are done, check your work against theanswers in the back of the book.1. Given the vectors v 1, 2 and w 2, 5 , find and sketch each vector.(a) v w(b) 3v(c) 2v 4w2. If possible, write w 2, 4, 1 as a linear combination of the vectors v1, v2, and v3.v1 1, 2, 0 ,v2 1, 0, 1 ,v3 0, 3, 0 3. Prove that the set of all singular 2 2 matrices is not a vector space.Historical EmphasisH ISTORICAL NOTEAugustin-Louis Cauchy(1789–1857)was encouraged by Pierre Simonde Laplace, one of France’s leading mathematicians, to studymathematics. Cauchy is oftencredited with bringing rigorto modern mathematics. Toread about his work, visitcollege.hmco.com/pic/larsonELA6e.NEW! Historical Notes are included throughout the text and feature brief biographiesof prominent mathematicians who contributed to linear algebra.Students are directed to the Web to read the full biographies, which are available viaPowerPoint Presentation.xi
Proven Pedagogy Integrated Technology Real-World ApplicationsComputer Algebra Systems and Graphing CalculatorsTechnologyNoteThe Technology Note feature in the text indicateshow students can utilize graphing calculators andcomputer algebra systems appropriately in theproblem-solving process.You can use a graphing utility or computer software program to find the unit vector for a givenvector. For example, you can use a graphing utility to find the unit vector for v 3, 4 , whichmay appear as:p gEXAMPLE 7NEW! Online Technology Guide provides the coveragestudents need to use computer algebra systems andgraphing calculators with this text.Provided on the accompanying student website, thisguide includes CAS and graphing calculator keystrokesfor select examples in the text. These examples featurean accompanying Technology Note, directing students tothe Guide for instruction on using their CAS/graphingcalculator to solve the example.In addition, the Guide provides an Introduction toMATLAB, Maple, Mathematica, and GraphingCalculators, as well as a section on Technology Pitfalls.Part I:I.1Using Elimination to Rewrite a System in Row-Echelon FormSolve the system.TechnologyNoteYou can use a computer softwareprogram or graphing utility witha built-in power regressionprogram to verify the result ofExample 10. For example, usingthe data in Table 5.2 and agraphing utility, a power fitprogram would result in ananswer of (or very similar to)y 1.00042x1.49954. Keystrokesand programming syntax forthese utilities/programs applicableto Example 10 are provided in theOnline Technology Guide,available at college.hmco.com/pic/larsonELA6e.Texas Instruments TI-83, TI-83 Plus, TI-84 Plus Graphing CalculatorSystems of Linear EquationsI.1.1 Basics: Press the ON key to begin using your TI-83 calculator. If you need to adjust the displaycontrast, first press 2nd, then press and hold(the up arrow key) to increase the contrast or(the downarrow key) to decrease the contrast. As you press and holdor, an integer between 0 (lightest) and9 (darkest) appears in the upper right corner of the display. When you have finished with the calculator, turnit off to conserve battery power by pressing 2nd and then OFF.Check the TI-83’s settings by pressing MODE. If necessary, use the arrow key to move the blinking cursorto a setting you want to change. Press ENTER to select a new setting. To start, select the options along theleft side of the MODE menu as illustrated in Figure I.1: normal display, floating display decimals, radianmeasure, function graphs, connected lines, sequential plotting, real number system, and full screen display.Details on alternative options will be given later in this guide. For now, leave the MODE menu by pressingCLEAR.x 2y 3z 9 x 3y 42x 5y 5z 17Keystrokes for TI-83Enter the system into matrix A.To rewrite the system in row-echelon form, use the following keystrokes.MATRX ALPHA [A] MATRX ENTER ENTERKeystrokes for TI-83 PlusEnter the system into matrix A.To rewrite the system in row-echelon form, use the following keystrokes.2nd [MATRX] ALPHA [A] 2nd [MATRX] ENTER ENTERKeystrokes for TI-84 PlusEnter the system into matrix A.To rewrite the system in row-echelon form, use the following keystrokes.2nd [MATRIX] ALPHA [A] 2nd [MATRIX] ENTER ENTERKeystrokes for TI-86Enter the system into matrix A.To rewrite the system in row-echelon form, use the following keystrokes.F4 ALPHA [A] ENTER2nd [MATRX] F4The Graphing Calculator Keystroke Guide offerscommands and instructions for various calculatorsand includes examples with step-by-step solutions,technology tips, and programs.The Graphing Calculator Keystroke Guide coversTI-83/TI-83 PLUS, TI-84 PLUS, TI-86, TI-89, TI-92,and Voyage 200.Also available on the student website: Electronic Data Sets are designed to be used with select exercises in the text and help students reinforceand broaden their technology skills using graphing calculators and computer algebra systems. MATLAB Exercises enhance students’ understanding of concepts using MATLAB software. Theseoptional exercises correlate to chapters in the text.xii
Additional Resources Get More from Your TextbookInstructor ResourcesStudent ResourcesInstructor Website This website offers instructors avariety of resources, including:Student Website This website offers comprehensive studyresources, including: NEW! Online Multimedia eBook NEW! Online Technology Guide Electronic Simulations MATLAB Exercises Graphing Calculator Keystroke Guide Chapters 8, 9, and 10 Electronic Data Sets Historical Note Biographies Instructor’s Solutions Manual, featuring completesolutions to all even-numbered exercises in the text. Digital Art and Figures, featuring key theoremsfrom the text.NEW! HM Testing (Powered by Diploma ) “Testingthe way you want it” HM Testing provides instructorswith a wide array of new algorithmic exercises along withimproved functionality and ease of use. Instructors cancreate, author/edit algorithmic questions, customize, anddeliver multiple types of tests.Student Solutions Manual Contains complete solutions toall odd-numbered exercises in the text.HM Math SPACE with Eduspace : Houghton Mifflin’s Online Learning Tool (powered by Blackboard )This web-based learning system provides instructors and students with powerful course management tools andtext-specific content to support all of their online teaching and learning needs. Eduspace now includes: NEW! WebAssign Developed by teachers, for teachers, WebAssign allows instructors to create assignments from anabundant ready-to-use database of algorithmic questions, or write and customize their own exercises. With WebAssign,instructors can: create, post, and review assignments 24 hours a day, 7 days a week; deliver, collect, grade, and recordassignments instantly; offer more practice exercises, quizzes and homework; assess student performance to keepabreast of individual progress; and capture the attention of online or distance-learning students. SMARTHINKING Live, Online Tutoring SMARTHINKING provides an easy-to-useand effective online, text-specific tutoring service. A dynamic Whiteboard and aGraphing Calculator function enable students and e-structors to collaborate easily.Online Course Content for Blackboard , WebCT , and eCollege Deliver program- or text-specific HoughtonMifflin content online using your institution’s local course management system. Houghton Mifflin offers homework andother resources formatted for Blackboard, WebCT, eCollege, and other course management systems. Add to an existingonline course or create a new one by selecting from a wide range of powerful learning and instructional materials.For more information, visit college.hmco.com/pic/larson/ELA6e or contact your local Houghton Mifflin sales representative.xiii
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What Is Linear Algebra?To answer the question “What is linear algebra?,” take a closer look at what you willstudy in this course. The most fundamental theme of linear algebra, and the first topiccovered in this textbook, is the theory of systems of linear equations. You have probablyencountered small systems of linear equations in your previous mathematics courses. Forexample, suppose you travel on an airplane between two cities that are 5000 kilometersapart. If the trip one way against a headwind takes 614 hours and the return trip the sameday in the direction of the wind takes only 5 hours, can you find the ground speed of theplane and the speed of the wind, assuming that both remain constant?If you let x represent the speed of the plane and y the speed of the wind, then thefollowing system models the problem.Original Flight6.25 x y 50005 x y 5000x yReturn FlightThis system of two equations and two unknowns simplifies tox y 800x y 1000,x yy1000x y 1000600(900, 100)200 200x2001000x y 800The lines intersect at (900, 100).and the solution is x 900 kilometers per hour and y 100 kilometers per hour.Geometrically, this system represents two lines in the xy-plane. You can see in the figurethat these lines intersect at the point 900, 100 , which verifies the answer that wasobtained.Solving systems of linear equations is one of the most important applications of linearalgebra. It has been argued that the majority of all mathematical problems encountered inscientific and i
We have designed Elementary Linear Algebra, Sixth Edition, for the introductory linear algebra course. Students embarking on a linear algebra course should have a thorough knowledge of algebra, and familiarity with analytic geometry and trigonometry. . lay an intuitive foundation for stu
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INTRODUCTION TO LINEAR ALGEBRA AND S-LINEAR ALGEBRA 1.1 Basic properties of linear algebra 7 1.2 Introduction to s-linear algebra 15 1.3 Some aapplications of S-linear algebra 30 Chapter Two INTRODUCTORY COCEPTS OF BASIC BISTRUCTURES AND S-BISTRUCTU
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MTH 210: Intro to Linear Algebra Fall 2019 Course Notes Drew Armstrong Linear algebra is the common denominator of modern mathematics. From the most pure to the most applied, if you use mathematics then you will use linear algebra. It is also a relatively new subject. Linear algebra as we
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