Elementary linear algebra : Metric version

468 Pages·2017·129.148 MB·English

by  Ron Larson

Elementary
N LINEAR
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ALGEBR A
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EIGHTH EDITION METRIC VERSION
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Ron Larson
Larson_9781337556217_se_Cov.indd 1 12/15/16 2:47 PM
INDEX OF APPLICATIONS
BIOLOGY AND LIFE SCIENCES Petroleum production, 292 MATHEMATICS AND GEOMETRY
Age distribution vector, 378, 391, 392, 395 Profit, from crops, 50 Adjoint of a matrix, 134, 135, 142, 146, 150
Age progression software, 180 Purchase of a product, 91 Collinear points in the xy-plane, 139, 143
Age transition matrix, 378, 391, 392, 395 Revenue Conic section(s), 226, 229
Agriculture, 37, 50 fast-food stand, 242 general equation, 141
Cosmetic surgery results simulation, 180 General Dynamics Corporation, 266, 276 rotation of axes, 221–224, 226, 229,
Duchenne muscular dystrophy, 365 Google, Inc., 291 383–385, 392, 395
Galloping speeds of animals, 276 telecommunications company, 242 Constrained optimization, 389, 390, 392,
Genetics, 365 software publishers, 143 395
Health care expenditures, 146 Sales, 37 Contraction in R2, 337, 341, 342
Heart rhythm analysis, 255 concession area, 42 Coplanar points in space, 140, 143
Hemophilia A, 365 stocks, 92 Cramer’s Rule, 130, 136, 137, 142, 143, 146
Hereditary baldness, 365 Wal-Mart, 32 Cross product of two vectors, 277–280,
Nutrition, 11 Sales promotion, 106 288, 289, 294
Population Satellite television service, 85, 86, 147 Differential equation(s)
of deer, 37 Software publishing, 143 linear, 218, 225, 226, 229
second order, 164
of laboratory mice, 91
system of first order, 354, 380, 381,
of rabbits, 379 ENGINEERING AND TECHNOLOGY
391, 392, 395, 396, 398
of sharks, 396
Aircraft design, 79 Expansion in R2, 337, 341, 342, 345
of small fish, 396
Circuit design, 322 Fibonacci sequence, 396
Population age and growth over time, 331
Computer graphics, 338 Fourier approximation(s), 285–287, 289, 292
Population genetics, 365
Computer monitors, 190 Geometry of linear transformations in R2,
Population growth, 378, 379, 391, 392,
Control system, 314 336–338, 341, 342, 345
395, 396, 398
Controllability matrix, 314 Hessian matrix, 375
Predator-prey relationship, 396
Cryptography, 94–96, 102, 107 Jacobian, 145
Red-green color blindness, 365
Data encryption, 94 Lagrange multiplier, 34
Reproduction rates of deer, 103
Decoding a message, 96, 102, 107 Laplace transform, 130
Sex-linked inheritance, 365 Digital signal processing, 172 Least squares approximation(s), 281–284, 289
Spread of a virus, 91, 93 Electrical network analysis, 30, 31, 34, 37, linear, 282, 289, 292
Vitamin C content, 11 quadratic, 283, 289, 292
150
Wound healing simulation, 180 Linear programming, 47
Electronic equipment, 190
X-linked inheritance, 365 Magnification in R2, 341, 342
Encoding a message, 95, 102, 107
Mathematical modeling, 273, 274, 276
Encryption key, 94
Parabola passing through three points, 150
BUSINESS AND ECONOMICS Engineering and control, 130
Partial fraction decomposition, 34, 37
Airplane allocation, 91 Error checking
Polynomial curve fitting, 25–28, 32, 34, 37
Borrowing money, 23 digit, 200
Quadratic form(s), 382–388, 392, 395, 398
Demand, for a rechargeable power drill, 103 matrix, 200
Quadric surface, rotation of, 388, 392
Demand matrix, external, 98 Feed horn, 223
Reflection in R2, 336, 341, 342, 345, 346
Economic system, 97, 98 Global Positioning System, 16
Relative maxima and minima, 375
of a small community, 103 Google’s Page Rank algorithm, 86
Rotation
Finance, 23 Image morphing and warping, 180 in R2, 303, 343, 393, 397
Fundraising, 92 Information retrieval, 58 in R3, 339, 340, 342, 345
Gasoline sales, 105 Internet search engine, 58 Second Partials Test for relative extrema, 375
Industrial system, 102, 107 Ladder network, 322 Shear in R2, 337, 338, 341, 342, 345
Input-output matrix, 97 Locating lost vessels at sea, 16 Taylor polynomial of degree 1, 282
Leontief input-output model(s), 97, 98, 103 Movie special effects, 180 Three-point form of the equation of a plane,
Major League Baseball salaries, 107 Network analysis, 29–34, 37 141, 143, 146
Manufacturing Radar, 172 Translation in R2, 308, 343
labor and material costs, 105 Sampling, 172 Triple scalar product, 288
models and prices, 150 Satellite dish, 223 Two-point form of the equation of a line,
production levels, 51, 105 Smart phones, 190 139, 143, 146, 150
Net profit, Microsoft, 32 Televisions, 190 Unit circle, 253
Output matrix, 98 Wireless communications, 172 Wronskian, 219, 225, 226, 229
Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-300
PHYSICAL SCIENCES Newton’s Second Law of Motion, 164 Smokers and nonsmokers, 91
Ohm’s Law, 322 Sports
Acoustical noise levels, 28
Pendulum, 225 activities, 91
Airplane speed, 11
Planetary periods, 27, 274 Super Bowl I, 36
Area
Primary additive colors, 190 Television watching, 91
of a parallelogram using cross product,
RGB color model, 190 Test scores, 108
279, 280, 288, 294
Stiffness matrix, 64, 72
of a triangle
Temperature, 34
using cross product, 289 STATISTICS AND PROBABILITY
Torque, 277
using determinants, 138, 142, 146,
Canonical regression analysis, 304
Traffic flow, 28, 33
150
Least squares regression
Undamped system, 164
Astronomy, 27, 274
analysis, 99–101, 103, 107, 265, 271–276
Unit cell, 213
Balancing a chemical equation, 4
cubic polynomial, 276
end-centered monoclinic, 213
Beam deflection, 64, 72
line, 100, 103, 107, 271, 274, 276, 296
Vertical motion, 37
Chemical
quadratic polynomial, 273, 276
Volume
changing state, 91
Leslie matrix, 331, 378
of a parallelepiped, 288, 289, 292
mixture, 37
Markov chain, 85, 86, 92, 93, 106
of a tetrahedron, 114, 140, 143
reaction, 4
absorbing, 89, 90, 92, 93, 106
Water flow, 33
Comet landing, 141
Multiple regression analysis, 304
Wind energy consumption, 103
Computational fluid dynamics, 79
Multivariate statistics, 304
Work, 248
Crystallography, 213
State matrix, 85, 106, 147, 331
Degree of freedom, 164
Steady state probability vector, 386
Diffusion, 354
SOCIAL SCIENCES AND Stochastic matrices, 84–86, 91–93, 106, 331
Dynamical systems, 396
DEMOGRAPHICS
Earthquake monitoring, 16
Electric and magnetic flux, 240 Caribbean Cruise, 106 MISCELLANEOUS
Flexibility matrix, 64, 72 Cellular phone subscribers, 107 Architecture, 388
Force Consumer preference model, 85, 86, 92, 147 Catedral Metropolitana Nossa Senhora
matrix, 72 Final grades, 105 Aparecida, 388
to pull an object up a ramp, 157 Grade distribution, 92 Chess tournament, 93
Geophysics, 172 Master’s degrees awarded, 276 Classified documents, 106
Grayscale, 190 Politics, voting apportionment, 51 Determining directions, 16
Hooke’s Law, 64 Population Dominoes, A2
Kepler’s First Law of Planetary Motion, 141 of consumers, 91 Flight crew scheduling, 47
Kirchhoff’s Laws, 30, 322 regions of the United States, 51 Sudoku, 120
Lattice of a crystal, 213 of smokers and nonsmokers, 91 Tips, 23
Mass-spring system, 164, 167 United States, 32 U.S. Postal Service, 200
Mean distance from the sun, 27, 274 world, 273 ZIP + 4 barcode, 200
Natural frequency, 164 Population migration, 106
Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-300
Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-300
Elementary Linear Algebra
8e Metric Version
Ron Larson
The Pennsylvania State University
The Behrend College
Australia • Brazil • Mexico • Singapore • United Kingdom • United States
Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-300
Elementary Linear Algebra, Eighth Edition, Metric Version © 2017 Cengage Learning
Ron Larson ALL RIGHTS RESERVED. No part of this work covered by the copyright
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Contents
1
Systems of Linear Equations 1
1.1 Introduction to Systems of Linear Equations 2
1.2 Gaussian Elimination and Gauss-Jordan Elimination 13
1.3 Applications of Systems of Linear Equations 25
Review Exercises 35
Project 1 Graphing Linear Equations 38
Project 2 Underdetermined and Overdetermined Systems 38
2
Matrices 39
2.1 Operations with Matrices 40
2.2 Properties of Matrix Operations 52
2.3 The Inverse of a Matrix 62
2.4 Elementary Matrices 74
2.5 Markov Chains 84
2.6 More Applications of Matrix Operations 94
Review Exercises 104
Project 1 Exploring Matrix Multiplication 108
Project 2 Nilpotent Matrices 108
3
Determinants 109
3.1 The Determinant of a Matrix 110
3.2 Determinants and Elementary Operations 118
3.3 Properties of Determinants 126
3.4 Applications of Determinants 134
Review Exercises 144
Project 1 Stochastic Matrices 147
Project 2 The Cayley-Hamilton Theorem 147
Cumulative Test for Chapters 1–3 149
4
Vector Spaces 151
4.1 Vectors in Rn 152
4.2 Vector Spaces 161
4.3 Subspaces of Vector Spaces 168
4.4 Spanning Sets and Linear Independence 175
4.5 Basis and Dimension 186
4.6 Rank of a Matrix and Systems of Linear Equations 195
4.7 Coordinates and Change of Basis 208
4.8 Applications of Vector Spaces 218
Review Exercises 227
Project 1 Solutions of Linear Systems 230
Project 2 Direct Sum 230
v
Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-300
vi Contents
5 Inner Product Spaces 231
5.1 Length and Dot Product in Rn 232
5.2 Inner Product Spaces 243
5.3 Orthonormal Bases: Gram-Schmidt Process 254
5.4 Mathematical Models and Least Squares Analysis 265
5.5 Applications of Inner Product Spaces 277
Review Exercises 290
Project 1 The QR-Factorization 293
Project 2 Orthogonal Matrices and Change of Basis 294
Cumulative Test for Chapters 4 and 5 295
6
Linear Transformations 297
6.1 Introduction to Linear Transformations 298
6.2 The Kernel and Range of a Linear Transformation 309
6.3 Matrices for Linear Transformations 320
6.4 Transition Matrices and Similarity 330
6.5 Applications of Linear Transformations 336
Review Exercises 343
Project 1 Reflections in R2 (I) 346
Project 2 Reflections in R2 (II) 346
7 Eigenvalues and Eigenvectors 347
7.1 Eigenvalues and Eigenvectors 348
7.2 Diagonalization 359
7.3 Symmetric Matrices and Orthogonal Diagonalization 368
7.4 Applications of Eigenvalues and Eigenvectors 378
Review Exercises 393
Project 1 Population Growth and Dynamical Systems (I) 396
Project 2 The Fibonacci Sequence 396
Cumulative Test for Chapters 6 and 7 397
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 1 The Mandelbrot Set
Project 2 Population Growth and Dynamical Systems (II)
Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-300
Contents vii
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 1 Beach Sand Replenishment (I)
Project 2 Beach Sand Replenishment (II)
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 1 The Successive Over-Relaxation (SOR) Method
Project 2 United States Population
Appendix A1
Mathematical Induction and Other Forms of Proofs
Answers to Odd-Numbered Exercises and Tests A7
Index A41
Technology Guide*
*Available online at CengageBrain.com.
Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-300
Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-300