Elementary Linear Algebra

565 Pages·2008·3.07 MB·English

by  Ron (Ron Larson) Larson

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
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Copyright ©2009 by Houghton Mifflin Harcourt Publishing Company.
All rights reserved.
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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
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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