ebook img

Elementary Linear Algebra PDF

565 Pages·2008·3.07 MB·English
Save to my drive
Quick download
Download
Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.

Preview Elementary Linear Algebra

INDEX OF APPLICATIONS BIOLOGY AND LIFE SCIENCES COMPUTERS AND COMPUTER SCIENCE Calories burned, 117 Computer graphics, 410–413, 415, 418 Population Computer operator, 142 of deer, 43 of rabbits, 459 ELECTRICAL ENGINEERING Population growth, 458–461, 472, 476, 477 Current flow in networks, 33, 36, 37, 40, 44 Reproduction rates of deer, 115 Kirchhoff’s Laws, 35, 36 Spread of a virus, 112 MATHEMATICS BUSINESS AND ECONOMICS Area of a triangle, 164, 169, 173 Average monthly cable television rates, 119 Collinear points, 165, 169 Basic cable and satellite television, 173 Conic sections and rotation, 265–270, 271–272, 275 Cable television service, 99, 101 Coplanar points, 167, 170 Consumer preference model, 99, 101, 174 Equation Consumer Price Index, 119 of a line, 165–166, 170, 174 Demand of a plane, 167–168, 170, 174 for a certain grade of gasoline, 115 Fourier approximations, 346–350, 351–352, 355 for a rechargeable power drill, 115 Linear differential equations in calculus, 262–265, Economic system, 107 270–271, 274–275 Industries, 114, 119 Quadratic forms, 463–471, 473, 476 Market research, 112 Systems of linear differential equations, 461–463, Net profit 472–473, 476 Microsoft, 38 Volume of a tetrahedron, 166, 170 Polo Ralph Lauren, 335 Number of stores MISCELLANEOUS Target Corporation, 354 Carbon dioxide emissions, 334 Production levels Cellular phone subscribers, 120 guitars, 59 College textbooks, 170 vehicles, 59 Doctorate degrees, 334 Profit from crops, 59 Fertilizer, 119 Retail sales of running shoes, 354 Final grades, 118 Revenue Flow eBay, Inc., 354 of traffic, 39, 40 Google, Inc., 354 of water, 39 Sales, 43 Gasoline, 117 Advanced Auto Parts, 334 Milk, 117 Auto Zone, 334 Motor vehicle registrations, 115 Circuit City Stores, 355 Network Dell, Inc., 335 of pipes, 39 Gateway, Inc., 334 of streets, 39, 40 Wal-Mart, 39 Subscribers of a cellular communications company, 170 Total cost of manufacturing, 59 Population, 118, 472, 476, 480 PHYSICAL SCIENCES of consumers, 112 Astronomy, 332 of smokers and nonsmokers, 112 Average monthly temperature, 43 of the United States, 38 Periods of planets, 31 Projected population of the United States, 173 World energy consumption, 354 Regional populations, 60 Television viewing, 112 SOCIAL AND BEHAVIORAL SCIENCES Voting population, 60 Sports World population, 330 average salaries of Major League Baseball players, 120 average salary for a National Football League player, NUMERICAL LINEAR ALGEBRA 354 Adjoint of a matrix, 158–160, 168–169, 173 basketball, 43 Cramer’s Rule, 161–163, 169–170, 173 Fiesta Bowl Championship Series, 41 Cross product of two vectors in space, 336–341, 350–351, Super Bowl I, 43 355 Super Bowl XLI, 41 Cryptography, 102, 113–114, 118–119 Test scores, 120–121 Geometry of linear transformations in the plane, 407–410, 413–414, 418 STATISTICS Idempotent matrix, 98 Least squares approximations, 341–346, 351, 355 Leontief input-output models, 105, 114, 119 Least squares regression analysis, 108, 114–115, 119–120 LU-factorization, 93–98, 116–117 QR-factorization, 356–357 Stochastic matrices, 98, 118 Elementary Linear Algebra SIXTH EDITION RON LARSON The Pennsylvania State University The Behrend College DAVID C. FALVO The Pennsylvania State University The Behrend College HOUGHTON MIFFLIN HARCOURT PUBLISHING COMPANY Boston New York Publisher:Richard Stratton Senior Sponsoring Editor:Cathy Cantin Senior Marketing Manager:Jennifer Jones Discipline Product Manager:Gretchen Rice King Associate Editor:Janine Tangney Associate Editor:Jeannine Lawless Senior Project Editor:Kerry Falvey Program Manager:Touraj Zadeh Senior Media Producer: Douglas Winicki Senior Content Manager:Maren Kunert Art and Design Manager: Jill Haber Cover Design Manager:Anne S. Katzeff Senior Photo Editor:Jennifer Meyer Dare Senior Composition Buyer:Chuck Dutton New Title Project Manager:Susan Peltier Manager of New Title Project Management:Pat O’Neill Editorial Assistant:Amy Haines Marketing Assistant:Michael Moore Editorial Assistant:Laura Collins Cover image: ©Carl Reader/age fotostock Copyright ©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 information storage or retrieval system without the prior written permission of Houghton Mifflin Harcourt Publishing Company unless such copying is expressly permitted by federal copyright law. Address inquiries to College Permissions, Houghton Mifflin Harcourt Publishing Company, 222 Berkeley Street, Boston, MA 02116-3764. Printed in the U.S.A. Library of Congress Control Number: 2007940572 Instructor’s examination copy ISBN-13: 978-0-547-00481-5 ISBN-10: 0-547-00481-8 For orders, use student text ISBNs ISBN-13: 978-0-618-78376-2 ISBN-10: 0-618-78376-8 123456789-DOC-12 11 10 09 08 Contents A WORD FROM THE AUTHORS vii WHAT IS LINEAR ALGEBRA? xv CHAPTER 1 SYSTEMS OF LINEAR EQUATIONS 1 1.1 Introduction to Systems of Linear Equations 1 1.2 Gaussian Elimination and Gauss-Jordan Elimination 14 1.3 Applications of Systems of Linear Equations 29 Review Exercises 41 Project 1 Graphing Linear Equations 44 Project 2 Underdetermined and Overdetermined Systems of Equations 45 CHAPTER 2 MATRICES 46 2.1 Operations with Matrices 46 2.2 Properties of Matrix Operations 61 2.3 The Inverse of a Matrix 73 2.4 Elementary Matrices 87 2.5 Applications of Matrix Operations 98 Review Exercises 115 Project 1 Exploring Matrix Multiplication 120 Project 2 Nilpotent Matrices 121 iii iv Contents CHAPTER 3 DETERMINANTS 122 3.1 The Determinant of a Matrix 122 3.2 Evaluation of a Determinant Using Elementary Operations 132 3.3 Properties of Determinants 142 3.4 Introduction to Eigenvalues 152 3.5 Applications of Determinants 158 Review Exercises 171 Project 1 Eigenvalues and Stochastic Matrices 174 Project 2 The Cayley-Hamilton Theorem 175 Cumulative Test for Chapters 1–3 177 CHAPTER 4 VECTOR SPACES 179 4.1 Vectors in Rn 179 4.2 Vector Spaces 191 4.3 Subspaces of Vector Spaces 198 4.4 Spanning Sets and Linear Independence 207 4.5 Basis and Dimension 221 4.6 Rank of a Matrix and Systems of Linear Equations 232 4.7 Coordinates and Change of Basis 249 4.8 Applications of Vector Spaces 262 Review Exercises 272 Project 1 Solutions of Linear Systems 275 Project 2 Direct Sum 276 CHAPTER 5 INNER PRODUCT SPACES 277 5.1 Length and Dot Product in Rn 277 5.2 Inner Product Spaces 292 5.3 Orthonormal Bases:Gram-Schmidt Process 306 5.4 Mathematical Models and Least Squares Analysis 320 5.5 Applications of Inner Product Spaces 336 Review Exercises 352 Project 1 The QR-Factorization 356 Project 2 Orthogonal Matrices and Change of Basis 357 Cumulative Test for Chapters 4 and 5 359 Contents v CHAPTER 6 LINEAR TRANSFORMATIONS 361 6.1 Introduction to Linear Transformations 361 6.2 The Kernel and Range of a Linear Transformation 374 6.3 Matrices for Linear Transformations 387 6.4 Transition Matrices and Similarity 399 6.5 Applications of Linear Transformations 407 Review Exercises 416 Project 1 Reflections in the Plane (I) 419 Project 2 Reflections in the Plane (II) 420 CHAPTER 7 EIGENVALUES AND EIGENVECTORS 421 7.1 Eigenvalues and Eigenvectors 421 7.2 Diagonalization 435 7.3 Symmetric Matrices and Orthogonal Diagonalization 446 7.4 Applications of Eigenvalues and Eigenvectors 458 Review Exercises 474 Project 1 Population Growth and Dynamical Systems (I) 477 Project 2 The Fibonacci Sequence 478 Cumulative Test for Chapters 6 and 7 479 CHAPTER 8 COMPLEX VECTOR SPACES (online)* 8.1 Complex Numbers 8.2 Conjugates and Division of Complex Numbers 8.3 Polar Form and DeMoivre's Theorem 8.4 Complex Vector Spaces and Inner Products 8.5 Unitary and Hermitian Matrices Review Exercises Project Population Growth and Dynamical Systems (II) vi Contents CHAPTER 9 LINEAR PROGRAMMING (online)* 9.1 Systems of Linear Inequalities 9.2 Linear Programming Involving Two Variables 9.3 The Simplex Method:Maximization 9.4 The Simplex Method:Minimization 9.5 The Simplex Method:Mixed Constraints Review Exercises Project Cholesterol Levels CHAPTER 10 NUMERICAL METHODS (online)* 10.1 Gaussian Elimination with Partial Pivoting 10.2 Iterative Methods for Solving Linear Systems 10.3 Power Method for Approximating Eigenvalues 10.4 Applications of Numerical Methods Review Exercises Project Population Growth APPENDIX MATHEMATICAL INDUCTION AND OTHER A1 FORMS OF PROOFS ONLINE TECHNOLOGY GUIDE (online)* ANSWER KEY A9 INDEX A59 *Available online at college.hmco.com/pic/larsonELA6e. A Word from the Authors Welcome! We have designed Elementary Linear Algebra, Sixth Edition, for the introductorylinear algebra course. Students embarking on a linear algebra course should have a thorough knowledge of algebra, and familiarity with analytic geometry and trigonometry. We do not assume that calculus is a prerequisite for this course, but we do include examples and exercises requir- ing 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 and concisely. To this end, we have carefully selected the examples and exercises to balance theory 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 and bases, beginning with a brief look leading into the vector space concept as a natural exten- sion of these familiar examples. This material is often the most difficult for students, but our approach to linear independence, span, basis, and dimension is carefully explained and illustrated by examples. The eigenvalue problem is developed in detail in Chapter 7, but we lay 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 program- ming, and numerical methods. They can be found on the student website for this text at college.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

Description:
The cornerstone of Elementary Linear Algebra is the authors' clear, careful, and concise presentation of material--written so that students can fully understand how mathematics works. This program balances theory with examples, applications, and geometric intuition for a complete, step-by-step learn
See more

The list of books you might like

Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.