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Features • Many fundamental ideas of linear algebra are introduced early, in the concrete setting of n , and then gradually examined from different points of view • Utilizing a modern view of matrix multiplication simplifies many arguments and ties vector space ideas into the study of linear systems • Every major concept is given a geometric interpretation to help students learn better by visualizing the idea • Keeping with the recommendations of the original LACSG, because orthogonality plays an important role in computer calculations and numerical linear algebra, and because inconsistent linear systems arise so often in practical work, this title includes a comprehensive treatment of both orthogonality and the least-squares problem • NEW! Reasonable Answers advice and exercises encourage students to ensure their computations are consistent with the data at hand and the questions being asked CVR_LAY1216_06_GE_CVR_Vivar.indd Lay • Lay • McDonald Available separately for purchase is MyLab Math for Linear Algebra and Its Applications, the teaching and learning platform that empowers instructors to personalize learning for every student When combined with Pearson’s trusted educational content, this optional suite helps deliver the learning outcomes desired This edition includes interactive versions of many of the figures in the text, letting students manipulate figures and experiment with matrices to gain a deeper geometric understanding of key concepts and principles SIXTH EDITION • Projects at the end of each chapter on a wide range of themes (including using linear transformations to create art and detecting and correcting errors in encoded messages) enhance student learning Linear Algebra and Its Applications Linear Algebra and Its Applications, now in its sixth edition, not only follows the recommendations of the original Linear Algebra Curriculum Study Group (LACSG) but also includes ideas currently being discussed by the LACSG 2.0 and continues to provide a modern elementary introduction to linear algebra This edition adds exciting new topics, examples, and online resources to highlight the linear algebraic foundations of machine learning, artificial intelligence, data science, and digital signal processing GLOBAL EDITION GLOB AL EDITION GLOBAL EDITION This is a special edition of an established title widely used by colleges and universities throughout the world Pearson published this exclusive edition for the benefit of students outside the United States and Canada If you purchased this book within the United States or Canada, you should be aware that it has been imported without the approval of the Publisher or Author Linear Algebra and Its Applications SIXTH EDITION David C Lay • Steven R Lay • Judi J McDonald 09/04/21 12:22 PM S I X T H E D I T I O N Linear Algebra and Its Applications G L O B A L E D I T I O N David C Lay University of Maryland–College Park Steven R Lay Lee University Judi J McDonald Washington State University Pearson Education Limited KAO Two KAO Park Hockham Way Harlow Essex CM17 9SR United Kingdom and Associated Companies throughout the world Visit us on the World Wide Web at: www.pearsonglobaleditions.com © Pearson Education Limited, 2022 The rights of David C Lay, Steven R Lay, and Judi J McDonald to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988 Authorized adaptation from the United States edition, entitled Linear Algebra and Its Applications, 6th Edition, ISBN 978-0-13-585125-8 by David C Lay, Steven R Lay, and Judi J McDonald, published by Pearson Education © 2021 All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without either the prior written permission of the publisher or a license permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Saffron House, 6–10 Kirby Street, London EC1N 8TS All trademarks used herein are the property of their respective owners The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any affiliation with or endorsement of this book by such owners For information regarding permissions, request forms, and the appropriate contacts within the Pearson Education Global Rights and Permissions department, please visit www.pearsoned.com/permissions This eBook is a standalone product and may or may not include all assets that were part of the print version It also does not provide access to other Pearson digital products like MyLab and Mastering The publisher reserves the right to remove any material in this eBook at any time British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN 10: 1-292-35121-7 ISBN 13: 978-1-292-35121-6 eBook ISBN 13: 978-1-292-35122-3 To my wife, Lillian, and our children, Christina, Deborah, and Melissa, whose support, encouragement, and faithful prayers made this book possible David C Lay About the Authors David C Lay As a founding member of the NSF-sponsored Linear Algebra Curriculum Study Group (LACSG), David Lay was a leader in the movement to modernize the linear algebra curriculum and shared those ideas with students and faculty through his authorship of the first four editions of this textbook David C Lay earned a B.A from Aurora University (Illinois), and an M.A and Ph.D from the University of California at Los Angeles David Lay was an educator and research mathematician for more than 40 years, mostly at the University of Maryland, College Park He also served as a visiting professor at the University of Amsterdam, the Free University in Amsterdam, and the University of Kaiserslautern, Germany He published more than 30 research articles on functional analysis and linear algebra Lay was also a coauthor of several mathematics texts, including Introduction to Functional Analysis with Angus E Taylor, Calculus and Its Applications, with L J Goldstein and D I Schneider, and Linear Algebra Gems—Assets for Undergraduate Mathematics, with D Carlson, C R Johnson, and A D Porter David Lay received four university awards for teaching excellence, including, in 1996, the title of Distinguished Scholar-Teacher of the University of Maryland In 1994, he was given one of the Mathematical Association of America’s Awards for Distinguished College or University Teaching of Mathematics He was elected by the university students to membership in Alpha Lambda Delta National Scholastic Honor Society and Golden Key National Honor Society In 1989, Aurora University conferred on him the Outstanding Alumnus award David Lay was a member of the American Mathematical Society, the Canadian Mathematical Society, the International Linear Algebra Society, the Mathematical Association of America, Sigma Xi, and the Society for Industrial and Applied Mathematics He also served several terms on the national board of the Association of Christians in the Mathematical Sciences In October 2018, David Lay passed away, but his legacy continues to benefit students of linear algebra as they study the subject in this widely acclaimed text About the Authors Steven R Lay Steven R Lay began his teaching career at Aurora University (Illinois) in 1971, after earning an M.A and a Ph.D in mathematics from the University of California at Los Angeles His career in mathematics was interrupted for eight years while serving as a missionary in Japan Upon his return to the States in 1998, he joined the mathematics faculty at Lee University (Tennessee) and has been there ever since Since then he has supported his brother David in refining and expanding the scope of this popular linear algebra text, including writing most of Chapters and Steven is also the author of three college-level mathematics texts: Convex Sets and Their Applications, Analysis with an Introduction to Proof, and Principles of Algebra In 1985, Steven received the Excellence in Teaching Award at Aurora University He and David, and their father, Dr L Clark Lay, are all distinguished mathematicians, and in 1989, they jointly received the Outstanding Alumnus award from their alma mater, Aurora University In 2006, Steven was honored to receive the Excellence in Scholarship Award at Lee University He is a member of the American Mathematical Society, the Mathematics Association of America, and the Association of Christians in the Mathematical Sciences Judi J McDonald Judi J McDonald became a co-author on the fifth edition, having worked closely with David on the fourth edition She holds a B.Sc in Mathematics from the University of Alberta, and an M.A and Ph.D from the University of Wisconsin As a professor of Mathematics, she has more than 40 publications in linear algebra research journals and more than 20 students have completed graduate degrees in linear algebra under her supervision She is an associate dean of the Graduate School at Washington State University and a former chair of the Faculty Senate She has worked with the mathematics outreach project Math Central (http://mathcentral.uregina.ca/) and is a member of the second Linear Algebra Curriculum Study Group (LACSG 2.0) Judi has received three teaching awards: two Inspiring Teaching awards at the University of Regina, and the Thomas Lutz College of Arts and Sciences Teaching Award at Washington State University She also received the College of Arts and Sciences Institutional Service Award at Washington State University Throughout her career, she has been an active member of the International Linear Algebra Society and the Association for Women in Mathematics She has also been a member of the Canadian Mathematical Society, the American Mathematical Society, the Mathematical Association of America, and the Society for Industrial and Applied Mathematics Contents About the Authors Preface 12 A Note to Students Chapter 22 Linear Equations in Linear Algebra 25 INTRODUCTORY EXAMPLE: Linear Models in Economics and Engineering 25 1.1 Systems of Linear Equations 26 1.2 Row Reduction and Echelon Forms 37 1.3 Vector Equations 50 1.4 The Matrix Equation Ax D b 61 1.5 Solution Sets of Linear Systems 69 1.6 Applications of Linear Systems 77 1.7 Linear Independence 84 1.8 Introduction to Linear Transformations 91 1.9 The Matrix of a Linear Transformation 99 1.10 Linear Models in Business, Science, and Engineering Projects 117 Supplementary Exercises 117 Chapter Matrix Algebra 109 121 INTRODUCTORY EXAMPLE: Computer Models in Aircraft Design 2.1 Matrix Operations 122 2.2 The Inverse of a Matrix 135 2.3 Characterizations of Invertible Matrices 145 2.4 Partitioned Matrices 150 2.5 Matrix Factorizations 156 2.6 The Leontief Input–Output Model 165 2.7 Applications to Computer Graphics 171 121 Contents 2.8 2.9 Chapter Subspaces of Rn 179 Dimension and Rank 186 Projects 193 Supplementary Exercises 193 Determinants 195 INTRODUCTORY EXAMPLE: Weighing Diamonds 195 3.1 Introduction to Determinants 196 3.2 Properties of Determinants 203 3.3 Cramer’s Rule, Volume, and Linear Transformations Projects 221 Supplementary Exercises 221 Chapter Vector Spaces 212 225 INTRODUCTORY EXAMPLE: Discrete-Time Signals and Digital Signal Processing 225 4.1 Vector Spaces and Subspaces 226 4.2 Null Spaces, Column Spaces, Row Spaces, and Linear Transformations 235 4.3 Linearly Independent Sets; Bases 246 4.4 Coordinate Systems 255 4.5 The Dimension of a Vector Space 265 4.6 Change of Basis 273 4.7 Digital Signal Processing 279 4.8 Applications to Difference Equations 286 Projects 295 Supplementary Exercises 295 Chapter Eigenvalues and Eigenvectors 297 INTRODUCTORY EXAMPLE: Dynamical Systems and Spotted Owls 5.1 Eigenvectors and Eigenvalues 298 5.2 The Characteristic Equation 306 5.3 Diagonalization 314 5.4 Eigenvectors and Linear Transformations 321 5.5 Complex Eigenvalues 328 5.6 Discrete Dynamical Systems 335 5.7 Applications to Differential Equations 345 5.8 Iterative Estimates for Eigenvalues 353 5.9 Applications to Markov Chains 359 Projects 369 Supplementary Exercises 369 297 Contents Chapter Orthogonality and Least Squares 373 INTRODUCTORY EXAMPLE: Artificial Intelligence and Machine Learning 373 6.1 Inner Product, Length, and Orthogonality 374 6.2 Orthogonal Sets 382 6.3 Orthogonal Projections 391 6.4 The Gram–Schmidt Process 400 6.5 Least-Squares Problems 406 6.6 Machine Learning and Linear Models 414 6.7 Inner Product Spaces 423 6.8 Applications of Inner Product Spaces 431 Projects 437 Supplementary Exercises 438 Chapter Symmetric Matrices and Quadratic Forms 441 INTRODUCTORY EXAMPLE: Multichannel Image Processing 441 7.1 Diagonalization of Symmetric Matrices 443 7.2 Quadratic Forms 449 7.3 Constrained Optimization 456 7.4 The Singular Value Decomposition 463 7.5 Applications to Image Processing and Statistics 473 Projects 481 Supplementary Exercises 481 Chapter The Geometry of Vector Spaces 483 INTRODUCTORY EXAMPLE: The Platonic Solids 8.1 8.2 8.3 8.4 8.5 8.6 Chapter Affine Combinations 484 Affine Independence 493 Convex Combinations 503 Hyperplanes 510 Polytopes 519 Curves and Surfaces 531 Project 542 Supplementary Exercises 543 Optimization 545 INTRODUCTORY EXAMPLE: The Berlin Airlift 9.1 9.2 9.3 9.4 483 545 Matrix Games 546 Linear Programming Geometric Method 560 Linear Programming Simplex Method 570 Duality 585 Project 594 Supplementary Exercises 594 Contents Chapter 10 Finite-State Markov Chains C-1 (Available Online) INTRODUCTORY EXAMPLE: Googling Markov Chains C-1 10.1 Introduction and Examples C-2 10.2 The Steady-State Vector and Google’s PageRank C-13 10.3 Communication Classes C-25 10.4 Classification of States and Periodicity C-33 10.5 The Fundamental Matrix C-42 10.6 Markov Chains and Baseball Statistics C-54 Appendixes A B Uniqueness of the Reduced Echelon Form Complex Numbers 599 604 Credits Glossary 605 Answers to Odd-Numbered Exercises Index I-1 A-1 597 Applications Index Biology and Ecology Estimating systolic blood pressure, 422 Laboratory animal trials, 367 Molecular modeling, 173–174 Net primary production of nutrients, 418–419 Nutrition problems, 109–111, 115 Predator-prey system, 336–337, 343 Spotted owls and stage-matrix models, 297–298, 341–343 Business and Economics Accelerator-multiplier model, 293 Average cost curve, 418–419 Car rental fleet, 116, 368 Cost vectors, 57 Equilibrium prices, 77–79, 82 Exchange table, 82 Feasible set, 460, 562 Gross domestic product, 170 Indifference curves, 460–461 Intermediate demand, 165 Investment, 294 Leontief exchange model, 25, 77–79 Leontief input–output model, 25, 165–171 Linear programming, 26, 111–112, 153, 484, 519, 522, 560–566 Loan amortization schedule, 293 Manufacturing operations, 57, 96 Marginal propensity to consume, 293 Markov chains, 311, 359–368, C-1–C-63 Maximizing utility subject to a budget constraint, 460–461 Population movement, 113, 115–116, 311, 361 Price equation, 170 Total cost curve, 419 Value added vector, 170 Variable cost model, 421 Computers and Computer Science Bézier curves and surfaces, 509, 531–532 CAD, 537, 541 Color monitors, 178 Computer graphics, 122, 171–177, 498–500 Cray supercomputer, 153 Data storage, 66, 163 Error-detecting and error-correcting codes, 447, 471 Game theory, 519 High-end computer graphics boards, 176 Homogeneous coordinates, 172–173, 174 Parallel processing, 25, 132 Perspective projections, 175–176 Vector pipeline architecture, 153 Virtual reality, 174 VLSI microchips, 150 Wire-frame models, 121, 171 Control Theory Controllable system, 296 Control systems engineering, 155 Decoupled system, 340, 346, 349 Deep space probe, 155 State-space model, 296, 335 Steady-state response, 335 Transfer function (matrix), 155 Electrical Engineering Branch and loop currents, 111–112 Circuit design, 26, 160 Current flow in networks, 111–112, 115–116 Discrete-time signals, 228, 279–280 Inductance-capacitance circuit, 242 Kirchhoff’s laws, 161 Ladder network, 161, 163–164 Laplace transforms, 155, 213 Linear filters, 287–288 Low-pass filter, 289, 413 Minimal realization, 162 Ohm’s law, 111–113, 161 RC circuit, 346–347 RLC circuit, 254 Page numbers denoted with “C” are found within the online chapter 10 Series and shunt circuits, 161 Transfer matrix, 161–162, 163 Engineering Aircraft performance, 422, 437 Boeing Blended Wing Body, 122 Cantilevered beam, 293 CFD and aircraft design, 121–122 Deflection of an elastic beam, 137, 144 Deformation of a material, 482 Equilibrium temperatures, 36, 116–117, 193 Feedback controls, 519 Flexibility and stiffness matrices, 137, 144 Heat conduction, 164 Image processing, 441–442, 473–474, 479 LU factorization and airflow, 122 Moving average filter, 293 Superposition principle, 95, 98, 112 Mathematics Area and volume, 195–196, 215–217 Attractors/repellers in a dynamical system, 338, 341, 343, 347, 351 Bessel’s inequality, 438 Best approximation in function spaces, 426–427 Cauchy-Schwarz inequality, 427 Conic sections and quadratic surfaces, 481 Differential equations, 242, 345–347 Fourier series, 434–436 Hermite polynomials, 272 Hypercube, 527–529 Interpolating polynomials, 49, 194 Isomorphism, 188, 260–261 Jacobian matrix, 338 Laguerre polynomials, 272 Laplace transforms, 155, 213 Legendre polynomials, 430 106 CHAPTER Linear Equations in Linear Algebra PROOF a By Theorem in Section 1.4, the columns of A span Rm if and only if for each b in Rm the equation Ax D b is consistent—in other words, if and only if for every b, the equation T x/ D b has at least one solution This is true if and only if T maps Rn onto Rm b The equations T x/ D and Ax D are the same except for notation So, by Theorem 11, T is one-to-one if and only if Ax D has only the trivial solution This happens if and only if the columns of A are linearly independent, as was already noted in the boxed statement (3) in Section 1.7 Statement (a) in Theorem 12 is equivalent to the statement “T maps Rn onto Rm if and only if every vector in Rm is a linear combination of the columns of A.” See Theorem in Section 1.4 In the next example and in some exercises that follow, column vectors are written in rows, such as x D x1 ; x2 /, and T x/ is written as T x1 ; x2 / instead of the more formal T x1 ; x2 // x2 EXAMPLE Let T x1 ; x2 / D 3x1 C x2 , 5x1 C 7x2 , x1 C 3x2 / Show that T is a one-to-one linear transformation Does T map R2 onto R3 ? e2 x1 e1 T x3 T SOLUTION When x and T x/ are written as column vectors, you can determine the standard matrix of T by inspection, visualizing the row–vector entry in Ax 3 3x1 C x2 ? ? Ä x T x/ D 5x1 C 7x2 D ? ? D x2 x1 C 3x2 ? ? computation of each Ä x 75 x2 (4) A a2 a1 x1 Span{a1, a2} The transformation T is not onto x2 So T is indeed a linear transformation, with its standard matrix A shown in (4) The columns of A are linearly independent because they are not multiples By Theorem 12(b), T is one-to-one To decide if T is onto R3 , examine the span of the columns of A Since A is 2, the columns of A span R3 if and only if A has pivot positions, by Theorem This is impossible, since A has only columns So the columns of A not span R3 , and the associated linear transformation is not onto R3 Practice Problems Let T W R2 ! R2 be the transformation that first performs a horizontal shear that maps e2 into e2 :5e1 (but leaves e1 unchanged) and then reflects the result through the x2 -axis Assuming that T is linear, find its standard matrix [Hint: Determine the final location of the images of e1 and e2 ] Suppose A is a matrix with pivots Let T x/ D Ax be a linear transformation from R5 into R7 Is T a one-to-one linear transformation? Is T onto R7 ? 1.9 Exercises In Exercises 1–10, assume that T is a linear transformation Find the standard matrix of T T W R3 ! R2 , T e1 / D 1; 3/, T e2 / D 4; 2/, and T e3 / D 5; 4/, where e1 , e2 , e3 are the columns of the 3 identity matrix T W R2 ! R4 , T e1 / D 2; 1; 2; 1/ and T e2 / D 5; 2; 0; 0/, where e1 D 1; 0/ and e2 D 0; 1/ T W R2 ! R2 rotates points (about the origin) through =2 radians (in the counterclockwise direction) 1.9 T W R2 ! R2 rotates points (about the origin) through =4 radians (since the number is negative, the actual rotation is p p clockwise) [Hint: T e1 / D 1= 2; 1= 2/.] ‹ 15 ‹ ‹ T W R2 ! R2 is a vertical shear transformation that maps e1 into e1 2e2 but leaves the vector e2 unchanged ‹ 16 ‹ ‹ T W R2 ! R2 is a horizontal shear transformation that leaves e1 unchanged and maps e2 into e2 C 5e1 2 T W R ! R first rotates points through =4 radians (since the number is negative, the actual rotation is clockwise) and then reflects ppointspthrough the horizontal x1 -axis [Hint: T e1 / D 1= 2; 1= 2/.] T W R2 ! R2 first reflects points through the vertical x2 -axis and then reflects points through the line x2 D x1 T W R2 ! R2 first performs a horizontal shear that transforms e2 into e2 3e1 (leaving e1 unchanged) and then reflects points through the line x2 D x1 10 T W R2 ! R2 first reflects points through the vertical x2 -axis and then rotates points =2 radians ‹ ‹ ‹ The Matrix of a Linear Transformation 107 32 3 ‹ x1 2x1 3x3 ‹ 54 x2 D 4x1 ‹ x3 x1 x2 C x3 ‹ Ä x ‹5 x2 ‹ x1 3x2 D 2x1 C x2 x1 In Exercises 17–20, show that T is a linear transformation by finding a matrix that implements the mapping Note that x1 ; x2 ; : : : are not vectors but are entries in vectors 17 T x1 ; x2 ; x3 ; x4 / D 0; x1 C x2 ; x2 C x3 ; x3 C x4 / 18 T x1 ; x2 / D 2x2 19 T x1 ; x2 ; x3 / D x1 3x1 ; x1 4x2 ; 0; x2 / 5x2 C 4x3 ; x2 20 T x1 ; x2 ; x3 ; x4 / D 2x1 C 3x3 4x4 6x3 / T W R4 ! R/ 21 Let T W R2 ! R2 be a linear transformation such that T x1 ; x2 / D x1 C x2 ; 4x1 C 5x2 / Find x such that T x/ D 3; 8/ 11 A linear transformation T W R2 ! R2 first reflects points through the x1 -axis and then reflects points through the x2 axis Show that T can also be described as a linear transformation that rotates points about the origin What is the angle of that rotation? 22 Let T W R2 ! R3 be a linear transformation such that T x1 ; x2 / D x1 2x2 ; x1 C 3x2 ; 3x1 2x2 / Find x such that T x/ D 1; 4; 9/ 12 Show that the transformation in Exercise is merely a rotation about the origin What is the angle of the rotation? 23 (T/F) A linear transformation T W Rn ! Rm is completely determined by its effect on the columns of the n n identity matrix 13 Let T W R2 ! R2 be the linear transformation such that T e1 / and T e2 / are the vectors shown in the figure Using the figure, sketch the vector T 2; 1/ x2 In Exercises 23–32, mark each statement True or False (T/F) Justify each answer 24 (T/F) A mapping T W Rn ! Rm is one-to-one if each vector in Rn maps onto a unique vector in Rm 25 (T/F) If T W R2 ! R2 rotates vectors about the origin through an angle , then T is a linear transformation T(e ) T(e1) x1 14 Let T W R2 ! R2 be a linear transformation with standard matrix A D Œa1 a2 , where a1 and a2 are Ä shown in the figure Using the figure, draw the image of under the transformation T x a2 x1 a1 In Exercises 15 and 16, fill in the missing entries of the matrix, assuming that the equation holds for all values of the variables 26 (T/F) The columns of the standard matrix for a linear transformation from Rn to Rm are the images of the columns of the n n identity matrix 27 (T/F) When two linear transformations are performed one after another, the combined effect may not always be a linear transformation 28 (T/F) Not every linear transformation from Rn to Rm is a matrix transformation 29 (T/F) A mapping T W Rn ! Rm is onto Rm if every vector x in Rn maps onto some vector in Rm 30 (T/F) The standard matrix of a linear transformation from R2 to R2 that reflects points through theÄhorizontal axis, the a vertical axis, or the origin has the form , where a d and d are ˙1 108 CHAPTER Linear Equations in Linear Algebra 31 (T/F) A is a matrix, then the transformation x 7! Ax cannot be one-to-one 32 (T/F) A is a matrix, then the transformation x 7! Ax cannot map R2 onto R3 In Exercises 33–36, determine if the specified linear transformation is (a) one-to-one and (b) onto Justify each answer 33 The transformation in Exercise 17 34 The transformation in Exercise 35 The transformation in Exercise 19 36 The transformation in Exercise 14 In Exercises 37 and 38, describe the possible echelon forms of the standard matrix for a linear transformation T Use the notation of Example in Section 1.2 37 T W R3 ! R4 is one-to-one 38 T W R4 ! R3 is onto 39 Let T W Rn ! Rm be a linear transformation, with A its standard matrix Complete the following statement to make it true: “T is one-to-one if and only if A has pivot columns.” Explain why the statement is true [Hint: Look in the exercises for Section 1.7.] 40 Let T W Rn ! Rm be a linear transformation, with A its standard matrix Complete the following statement to make it true: “T maps Rn onto Rm if and only if A has pivot columns.” Find some theorems that explain why the statement is true 41 Verify the uniqueness of A in Theorem 10 Let T W Rn ! Rm be a linear transformation such that T x/ D B x for some m n matrix B Show that if A is the standard matrix for T , then A D B [Hint: Show that A and B have the same columns.] 42 Why is the question “Is the linear transformation T onto?” an existence question? 43 If a linear transformation T W Rn ! Rm maps Rn onto Rm , can you give a relation between m and n? If T is one-to-one, what can you say about m and n? 44 Let S W Rp ! Rn and T W Rn ! Rm be linear transformations Show that the mapping x 7! T S.x// is a linear transformation (from Rp to Rm ) [Hint: Compute T S.c u C d v// for u; v in Rp and scalars c and d Justify each step of the computation, and explain why this computation gives the desired conclusion.] T In Exercises 45–48, let T be the linear transformation whose standard matrix is given In Exercises 45 and 46, decide if T is a one-to-one mapping In Exercises 47 and 48, decide if T maps R5 onto R5 Justify your answers 3 10 6 77 10 16 47 45 46 4 12 12 75 7 6 12 87 14 47 10 65 6 3 13 6 14 15 47 12 97 48 6 85 13 14 15 11 STUDY GUIDE offers additional resources for mastering existence and uniqueness Solution to Practice Problems Follow what happens to e1 and e2 See Figure First, e1 is unaffected by the shear and then is reflected into e1 So T e1 / D e1 Second, e2 goes to e2 :5e1 by the shear transformation Since reflection through the x2 -axis changes e1 into e1 and leaves e2 unchanged, the vector e2 :5e1 goes to e2 C :5e1 So T e2 / D e2 C :5e1 x2 x2 x2 2.5 1 x1 x1 Shear transformation FIGURE The composition of two transformations 21 x1 Reflection through the x2-axis 1.10 Linear Models in Business, Science, and Engineering 109 Thus the standard matrix of T is T e1 / T e2 / D e1 e2 C :5e1 D Ä :5 The standard matrix representation of T is the matrix A Since A has columns and pivots, there is a pivot in every column so the columns are linearly independent By Theorem 12, T is one-to-one Since A has rows and only pivots, there is not a pivot in every row hence the columns of A not span R7 By Theorem 12, and T is not onto 1.10 Linear Models in Business, Science, and Engineering The mathematical models in this section are all linear; that is, each describes a problem by means of a linear equation, usually in vector or matrix form The first model concerns nutrition but actually is representative of a general technique in linear programming problems The second model comes from electrical engineering The third model introduces the concept of a linear difference equation, a powerful mathematical tool for studying dynamic processes in a wide variety of fields such as engineering, ecology, economics, telecommunications, and the management sciences Linear models are important because natural phenomena are often linear or nearly linear when the variables involved are held within reasonable bounds Also, linear models are more easily adapted for computer calculation than are complex nonlinear models As you read about each model, pay attention to how its linearity reflects some property of the system being modeled Constructing a Nutritious Weight-Loss Diet The formula for the Cambridge Diet, a popular diet in the 1980s, was based on years of research A team of scientists headed by Dr Alan H Howard developed this diet at Cambridge University after more than eight years of clinical work with obese patients.1 The very low-calorie powdered formula diet combines a precise balance of carbohydrate, high-quality protein, and fat, together with vitamins, minerals, trace elements, and electrolytes Millions of persons have used the diet to achieve rapid and substantial weight loss To achieve the desired amounts and proportions of nutrients, Dr Howard had to incorporate a large variety of foodstuffs in the diet Each foodstuff supplied several of the required ingredients, but not in the correct proportions For instance, nonfat milk was a major source of protein but contained too much calcium So soy flour was used for part of the protein because soy flour contains little calcium However, soy flour contains proportionally too much fat, so whey was added since it supplies less fat in relation to calcium Unfortunately, whey contains too much carbohydrate: : : : The following example illustrates the problem on a small scale Listed in Table are three of the ingredients in the diet, together with the amounts of certain nutrients supplied by 100 grams (g) of each ingredient.2 The first announcement of this rapid weight-loss regimen was given in the International Journal of Obesity (1978) 2, 321–332 Ingredients in the diet as of 1984; nutrient data for ingredients adapted from USDA Agricultural Handbooks No 8-1 and 8-6, 1976 110 CHAPTER Linear Equations in Linear Algebra TABLE The Cambridge Diet Amounts (g) Supplied per 100 g of Ingredient Nonfat milk Soy flour Whey Amounts (g) Supplied by Cambridge Diet in One Day Protein 36 51 13 33 Carbohydrate 52 34 74 45 Nutrient Fat 1.1 EXAMPLE If possible, find some combination of nonfat milk, soy flour, and whey to provide the exact amounts of protein, carbohydrate, and fat supplied by the diet in one day (Table 1) SOLUTION Let x1 , x2 , and x3 , respectively, denote the number of units (100 g) of these foodstuffs One approach to the problem is to derive equations for each nutrient separately For instance, the product x1 units of nonfat milk protein per unit of nonfat milk gives the amount of protein supplied by x1 units of nonfat milk To this amount, we would then add similar products for soy flour and whey and set the resulting sum equal to the amount of protein we need Analogous calculations would have to be made for each nutrient A more efficient method, and one that is conceptually simpler, is to consider a “nutrient vector” for each foodstuff and build just one vector equation The amount of nutrients supplied by x1 units of nonfat milk is the scalar multiple Scalar x1 units of nonfat milk Vector nutrients per unit D x1 a1 of nonfat milk (1) where a1 is the first column in Table Let a2 and a3 be the corresponding vectors for soy flour and whey, respectively, and let b be the vector that lists the total nutrients required (the last column of the table) Then x2 a2 and x3 a3 give the nutrients supplied by x2 units of soy flour and x3 units of whey, respectively So the relevant equation is x1 a1 C x2 a2 C x3 a3 D b (2) Row reduction of the augmented matrix for the corresponding system of equations shows that 3 36 51 13 33 0 :277 52 34 74 45 :392 1.1 0 :233 To three significant digits, the diet requires 277 units of nonfat milk, 392 units of soy flour, and 233 units of whey in order to provide the desired amounts of protein, carbohydrate, and fat It is important that the values of x1 , x2 , and x3 found above are nonnegative This is necessary for the solution to be physically feasible (How could you use :233 units of whey, for instance?) With a large number of nutrient requirements, it may be necessary to use a larger number of foodstuffs in order to produce a system of equations with a 1.10 Linear Models in Business, Science, and Engineering 111 “nonnegative” solution Thus many, many different combinations of foodstuffs may need to be examined in order to find a system of equations with such a solution In fact, the manufacturer of the Cambridge Diet was able to supply 31 nutrients in precise amounts using only 33 ingredients The diet construction problem leads to the linear equation (2) because the amount of nutrients supplied by each foodstuff can be written as a scalar multiple of a vector, as in (1) That is, the nutrients supplied by a foodstuff are proportional to the amount of the foodstuff added to the diet mixture Also, each nutrient in the mixture is the sum of the amounts from the various foodstuffs Problems of formulating specialized diets for humans and livestock occur frequently Usually they are treated by linear programming techniques Our method of constructing vector equations often simplifies the task of formulating such problems Linear Equations and Electrical Networks Current flow in a simple electrical network can be described by a system of linear equations A voltage source such as a battery forces a current of electrons to flow through the network When the current passes through a resistor (such as a lightbulb or motor), some of the voltage is “used up”; by Ohm’s law, this “voltage drop” across a resistor is given by V D RI where the voltage V is measured in volts, the resistance R in ohms (denoted by ), and the current flow I in amperes (amps, for short) The network in Figure contains three closed loops The currents flowing in loops 1, 2, and are denoted by I1 ; I2 , and I3 , respectively The designated directions of such loop currents are arbitrary If a current turns out to be negative, then the actual direction of current flow is opposite to that chosen in the figure If the current direction shown is away from the positive (longer) side of a battery ( ) around to the negative (shorter) side, the voltage is positive; otherwise, the voltage is negative Current flow in a loop is governed by the following rule KIRCHHOFF’S VOLTAGE LAW The algebraic sum of the RI voltage drops in one direction around a loop equals the algebraic sum of the voltage sources in the same direction around the loop EXAMPLE Determine the loop currents in the network in Figure 30 volts 4V A 4V I1 B 3V 1V C 1V volts 1V I3 20 volts FIGURE 1V I2 D 1V SOLUTION For loop 1, the current I1 flows through three resistors, and the sum of the RI voltage drops is 4I1 C 4I1 C 3I1 D C C 3/I1 D 11I1 Current from loop also flows in part of loop 1, through the short branch between A and B The associated RI drop there is 3I2 volts However, the current direction for the branch AB in loop is opposite to that chosen for the flow in loop 2, so the algebraic sum of all RI drops for loop is 11I1 3I2 Since the voltage in loop is C30 volts, Kirchhoff’s voltage law implies that 11I1 3I2 D 30 112 CHAPTER Linear Equations in Linear Algebra The equation for loop is 3I1 C 6I2 I3 D The term 3I1 comes from the flow of the loop current through the branch AB (with a negative voltage drop because the current flow there is opposite to the flow in loop 2) The term 6I2 is the sum of all resistances in loop 2, multiplied by the loop current The term I3 D I3 comes from the loop current flowing through the 1-ohm resistor in branch CD, in the direction opposite to the flow in loop The loop equation is I2 C 3I3 D 25 Note that the 5-volt battery in branch CD is counted as part of both loop and loop 3, but it is volts for loop because of the direction chosen for the current in loop The 20-volt battery is negative for the same reason The loop currents are found by solving the system 11I1 3I2 D 3I1 C 6I2 I3 D I2 C 3I3 D 30 25 (3) Row operations on the augmented matrix lead to the solution: I1 D amps, I2 D amp, and I3 D amps The negative value of I3 indicates that the actual current in loop flows in the direction opposite to that shown in Figure It is instructive to look at system (3) as a vector equation: 3 3 11 30 I1 C I2 C I3 D 5 25 ✻ r1 ✻ r2 ✻ r3 (4) ✻ v The first entry of each vector concerns the first loop, and similarly for the second and third entries The first resistor vector r1 lists the resistance in the various loops through which current I1 flows A resistance is written negatively when I1 flows against the flow direction in another loop Examine Figure and see how to compute the entries in r1 ; then the same for r2 and r3 The matrix form of equation (4), I1 Ri D v; where R D Œ r1 r2 r3 and i D I2 I3 provides a matrix version of Ohm’s law If all loop currents are chosen in the same direction (say, counterclockwise), then all entries off the main diagonal of R will be negative The matrix equation Ri D v makes the linearity of this model easy to see at a glance For instance, if the voltage vector is doubled, then the current vector must double Also, a superposition principle holds That is, the solution of equation (4) is the sum of the solutions of the equations 3 30 0 Ri D 5; Ri D 5; and Ri D 0 25 1.10 Linear Models in Business, Science, and Engineering 113 Each equation here corresponds to the circuit with only one voltage source (the other sources being replaced by wires that close each loop) The model for current flow is linear precisely because Ohm’s law and Kirchhoff’s law are linear: The voltage drop across a resistor is proportional to the current flowing through it (Ohm), and the sum of the voltage drops in a loop equals the sum of the voltage sources in the loop (Kirchhoff) Loop currents in a network can be used to determine the current in any branch of the network If only one loop current passes through a branch, such as from B to D in Figure 1, the branch current equals the loop current If more than one loop current passes through a branch, such as from A to B , the branch current is the algebraic sum of the loop currents in the branch (Kirchhoff’s current law) For instance, the current in branch AB is I1 I2 D D amps, in the direction of I1 The current in branch CD is I2 I3 D amps Difference Equations In many fields, such as ecology, economics, and engineering, a need arises to model mathematically a dynamic system that changes over time Several features of the system are each measured at discrete time intervals, producing a sequence of vectors x0 , x1 , x2 ; : : : : The entries in xk provide information about the state of the system at the time of the k th measurement If there is a matrix A such that x1 D Ax0 , x2 D Ax1 , and, in general, xk C1 D Axk for k D 0; 1; 2; : : : (5) then (5) is called a linear difference equation (or recurrence relation) Given such an equation, one can compute x1 , x2 , and so on, provided x0 is known Sections 4.8 and several sections in Chapter will develop formulas for xk and describe what can happen to xk as k increases indefinitely The discussion below illustrates how a difference equation might arise A subject of interest to demographers is the movement of populations or groups of people from one region to another The simple model here considers the changes in the population of a certain city and its surrounding suburbs over a period of years Fix an initial year—say, 2020—and denote the populations of the city and suburbs that year by r0 and s0 , respectively Let x0 be the population vector Ä r City population, 2020 x0 D s0 Suburban population, 2020 For 2021 and subsequent years, denote the populations of the city and suburbs by the vectors Ä Ä Ä r1 r2 r x1 D ; x2 D ; x3 D ; : : : s1 s2 s3 Our goal is to describe mathematically how these vectors might be related Suppose demographic studies show that each year about 5% of the city’s population moves to the suburbs (and 95% remains in the city), while 3% of the suburban population moves to the city (and 97% remains in the suburbs) See Figure After year, the original r0 persons in the city are now distributed between city and suburbs as Ä Ä :95r0 :95 Remain in city D r0 (6) :05r0 :05 Move to suburbs The s0 persons in the suburbs in 2020 are distributed year later as Ä :03 Move to city s0 :97 Remain in suburbs (7) 114 CHAPTER Linear Equations in Linear Algebra City Suburbs 05 95 97 03 FIGURE Annual percentage migration between city and suburbs The vectors in (6) and (7) account for all of the population in 2021.3 Thus Ä Ä Ä Ä Ä r1 :95 :03 :95 :03 r0 D r0 C s0 D s1 :05 :97 :05 :97 s0 That is, x1 D M x0 (8) where M is the migration matrix determined by the following table: Ä From: City Suburbs To: :95 :05 City Suburbs :03 :97 Equation (8) describes how the population changes from 2020 to 2021 If the migration percentages remain constant, then the change from 2021 to 2022 is given by x2 D M x1 and similarly for 2022 to 2023 and subsequent years In general, xk C1 D M xk for k D 0; 1; 2; : : : (9) The sequence of vectors fx0 ; x1 ; x2 ; : : :g describes the population of the city/suburban region over a period of years EXAMPLE Compute the population of the region just described for the years 2021 and 2022, given that the population in 2020 was 600,000 in the city and 400,000 in the suburbs Ä 600;000 SOLUTION The initial population in 2020 is x0 D For 2021, 400;000 x1 D Ä :95 :05 For 2022, x2 D M x1 D Ä :03 :97 :95 :05 Ä 600;000 400;000 :03 :97 Ä D 582;000 418;000 Ä 582;000 418;000 D Ä 565;440 434;560 For simplicity, we ignore other influences on the population such as births, deaths, and migration into and out of the city/suburban region 1.10 Linear Models in Business, Science, and Engineering 115 The model for population movement in (9) is linear because the correspondence xk 7! xk C1 is a linear transformation The linearity depends on two facts: the number of people who chose to move from one area to another is proportional to the number of people in that area, as shown in (6) and (7), and the cumulative effect of these choices is found by adding the movement of people from the different areas Practice Problem Find a matrix A and vectors x and b such that the problem in Example amounts to solving the equation Ax D b 1.10 Exercises The container of a breakfast cereal usually lists the number of calories and the amounts of protein, carbohydrate, and fat contained in one serving of the cereal The amounts for two common cereals are given below Suppose a mixture of these two cereals is to be prepared that contains exactly 295 calories, g of protein, 48 g of carbohydrate, and g of fat a Set up a vector equation for this problem Include a statement of what the variables in your equation represent b Write an equivalent matrix equation, and then determine if the desired mixture of the two cereals can be prepared Nutrition Information per Serving General Mills Quaker® Nutrient Cheerios® 100% Natural Cereal Calories 110 130 Protein (g) Carbohydrate (g) 20 18 Fat (g) One serving of Post Shredded Wheat® supplies 160 calories, g of protein, g of fiber, and g of fat One serving of Crispix® supplies 110 calories, g of protein, g of fiber, and g of fat a Set up a matrix B and a vector u such that B u gives the amounts of calories, protein, fiber, and fat contained in a mixture of three servings of Shredded Wheat and two servings of Crispix T b Suppose that you want a cereal with more fiber than Crispix but fewer calories than Shredded Wheat Is it possible for a mixture of the two cereals to supply 130 calories, 3.20 g of protein, 2.46 g of fiber, and 64 g of fat? If so, what is the mixture? After taking a nutrition class, a big Annie’s® Mac and Cheese fan decides to improve the levels of protein and fiber in her favorite lunch by adding broccoli and canned chicken The nutritional information for the foods referred to in this are given in the table Nutrition Information per Serving Nutrient Mac and Cheese Broccoli Chicken Shells Calories 270 51 70 260 Protein (g) 10 5.4 15 Fiber (g) 5.2 T a If she wants to limit her lunch to 400 calories but get 30 g of protein and 10 g of fiber, what proportions of servings of Mac and Cheese, broccoli, and chicken should she use? T b She found that there was too much broccoli in the propor- tions from part (a), so she decided to switch from classical Mac and Cheese to Annie’s® Whole Wheat Shells and White Cheddar What proportions of servings of each food should she use to meet the same goals as in part (a)? The Cambridge Diet supplies g of calcium per day, in addition to the nutrients listed in Table for Example The amounts of calcium per unit (100 g) supplied by the three ingredients in the Cambridge Diet are as follows: 1.26 g from nonfat milk, 19 g from soy flour, and g from whey Another ingredient in the diet mixture is isolated soy protein, which provides the following nutrients in each unit: 80 g of protein, g of carbohydrate, 3.4 g of fat, and 18 g of calcium a Set up a matrix equation whose solution determines the amounts of nonfat milk, soy flour, whey, and isolated soy protein necessary to supply the precise amounts of protein, carbohydrate, fat, and calcium in the Cambridge Diet State what the variables in the equation represent T b Solve the equation in (a) and discuss your answer T In Exercises 5–8, write a matrix equation that determines the loop currents If MATLAB or another matrix program is available, solve the system for the loop currents 116 Linear Equations in Linear Algebra CHAPTER 20 V 1V 1V I3 1V I4 2V 5V 6V I2 2V Returned To: Airport East West 500 cars, at three locations A car rented at one location may be returned to any of the three locations The various fractions of cars returned to the three locations are shown in the matrix below Suppose that on Monday there are 295 cars at the airport (or rented from there), 55 cars at the east side office, and 150 cars at the west side office What will be the approximate distribution of cars on Wednesday? 20 V 50 V trucks in Pullman, Spokane, and Seattle, respectively A truck rented at one location may be returned to any of the three locations The various fractions of trucks returned to the three locations each month are shown in the matrix below What will be the approximate distribution of the trucks after three months? T 12 Budget® Rent a Car in Wichita, Kansas, has a fleet of about 3V I3 10 In a certain region, about 6% of a city’s population moves to the surrounding suburbs each year, and about 4% of the suburban population moves into the city In 2020, there were 10,000,000 residents in the city and 800,000 in the suburbs Set up a difference equation that describes this situation, where x0 is the initial population in 2020 Then estimate the populations in the city and in the suburbs two years later, in 2022 Trucks Rented From: Pullman Spokane Seattle :30 :15 :05 4:30 :70 :055 :40 :15 :90 10 V I4 4V the populations in the city and in the suburbs two years later, in 2022 (Ignore other factors that might influence the population sizes.) T 11 College Moving Truck Rental has a fleet of 20, 100, and 200 4V I1 4V 3V 4V 7V 30 V 40 V 10 V 3V 40 V 1V I2 2V 20 V I4 2V 4V 2V I3 10 V 20 V 1V 2V 1V I1 1V 30 V I2 10 V 4V 30 V 4V I1 5V 3V 3V 1V 40 V 1V I1 3V I4 4V 1V 2V 3V I5 3V 1V 1V Returned To: Airport East West 3V I2 I3 2V 2V 30 V Cars Rented From: Airport East West :97 :05 :10 4:00 :90 :055 :03 :05 :85 20 V T 13 Let M and x0 be as in Example a Compute the population vectors xk for k D 1; : : : ; 20 Discuss what you find b Repeat part (a) with an initial population of 350,000 in the city and 650,000 in the suburbs What you find? T 14 Study how changes in boundary temperatures on a steel plate In a certain region, about 7% of a city’s population moves to the surrounding suburbs each year, and about 5% of the suburban population moves into the city In 2020, there were 800,000 residents in the city and 500,000 in the suburbs Set up a difference equation that describes this situation, where x0 is the initial population in 2020 Then estimate affect the temperatures at interior points on the plate a Begin by estimating the temperatures T1 , T2 , T3 , T4 at each of the sets of four points on the steel plate shown in the figure In each case, the value of Tk is approximated by the average of the temperatures at the four closest points See Exercises 43 and 44 in Section 1.1, where the values Supplementary Exercises Chapter (in degrees) turn out to be 20; 27:5; 30; 22:5/ How is this list of values related to your results for the points in set (a) and set (b)? b Without making any computations, guess the interior temperatures in (a) when the boundary temperatures are all multiplied by Check your guess 08 08 c Finally, make a general conjecture about the correspondence from the list of eight boundary temperatures to the list of four interior temperatures Plate A Plate B 208 208 08 08 2 4 208 208 108 108 08 08 108 108 (a) Solution to Practice Problem 36 51 13 A D 52 34 74 5; 1:1 117 408 408 (b) x1 x D x2 5; x3 33 b D 45 CHAPTER PROJECTS Chapter projects are available online A Interpolating Polynomials: This project shows how to use a system of linear equations to fit a polynomial through a set of points B Splines: This project also shows how to use a system of linear equations to fit a piecewise polynomial curve through a set of points D The Art of Linear Transformations: In this project, it is illustrated how to graph a polygon and then use linear transformations to change its shape and create a design E Loop Currents: The purpose of this project is to provide more and larger examples of loop currents F Diet: The purpose of this project is to provide examples of vector equations that result from balancing nutrients in a diet C Network Flows: The purpose of this project is to show how systems of linear equations may be used to model flow through a network CHAPTER SUPPLEMENTARY EXERCISES Mark each statement True or False (T/ F) Justify each answer (If true, cite appropriate facts or theorems If false, explain why or give a counterexample that shows why the statement is not true in every case (T/F) Every matrix is row equivalent to a unique matrix in echelon form (T/F) If a system Ax D b has more than one solution, then so does the system Ax D (T/F) If A is an m n matrix and the equation Ax D b is consistent for some b, then the columns of A span Rm (T/F) Any system of n linear equations in n variables has at most n solutions (T/F) If an augmented matrix Œ A b can be transformed by elementary row operations into reduced echelon form, then the equation Ax D b is consistent (T/F) If a system of linear equations has two different solutions, it must have infinitely many solutions (T/F) If matrices A and B are row equivalent, they have the same reduced echelon form (T/F) If a system of linear equations has no free variables, then it has a unique solution (T/F) If an augmented matrix Œ A b is transformed into Œ C d by elementary row operations, then the equations Ax D b and C x D d have exactly the same solution sets 10 (T/F) The equation Ax D has the trivial solution if and only if there are no free variables 11 (T/F) If A is an m n matrix and the equation Ax D b is consistent for every b in Rm , then A has m pivot columns 118 CHAPTER Linear Equations in Linear Algebra 12 (T/F) If an m n matrix A has a pivot position in every row, then the equation Ax D b has a unique solution for each b in Rm no points in common Typical graphs are illustrated in the figure 13 (T/F) If an n n matrix A has n pivot positions, then the reduced echelon form of A is the n n identity matrix 14 (T/F) If 3 matrices A and B each have three pivot positions, then A can be transformed into B by elementary row operations 15 (T/F) If A is an m n matrix, if the equation Ax D b has at least two different solutions, and if the equation Ax D c is consistent, then the equation Ax D c has many solutions Three planes intersecting in a line (a) Three planes intersecting in a point (b) Three planes with no intersection (c) Three planes with no intersection (c') 16 (T/F) If A and B are row equivalent m n matrices and if the columns of A span Rm , then so the columns of B 17 (T/F) If none of the vectors in the set S D fv1 ; v2 ; v3 g in R3 is a multiple of one of the other vectors, then S is linearly independent 18 (T/F) If fu; v; wg is linearly independent, then u, v, and w are not in R2 19 (T/F) In some cases, it is possible for four vectors to span R5 20 (T/F) If u and v are in Rm , then u is in Spanfu; vg 21 (T/F) If u, v, and w are nonzero vectors in R2 , then w is a linear combination of u and v 22 (T/F) If w is a linear combination of u and v in Rn , then u is a linear combination of v and w 23 (T/F) Suppose that v1 , v2 , and v3 are in R , v2 is not a multiple of v1 , and v3 is not a linear combination of v1 and v2 Then fv1 ; v2 ; v3 g is linearly independent 28 Suppose the coefficient matrix of a linear system of three equations in three variables has a pivot position in each column Explain why the system has a unique solution 29 Determine h and k such that the solution set of the system (i) is empty, (ii) contains a unique solution, and (iii) contains infinitely many solutions x1 C 3x2 D k b 2x1 C hx2 D a 4x1 C hx2 D 6x1 C kx2 D 30 Consider the problem of determining whether the following system of equations is consistent: 4x1 8x1 2x2 C 7x3 D 3x2 C 10x3 D 24 (T/F) A linear transformation is a function a Define appropriate vectors, and restate the problem in terms of linear combinations Then solve that problem 25 (T/F) If A is a matrix, the linear transformation x 7! Ax cannot map R5 onto R6 b Define an appropriate matrix, and restate the problem using the phrase “columns of A.” 26 Let a and b represent real numbers Describe the possible solution sets of the (linear) equation ax D b [Hint: The number of solutions depends upon a and b ] 27 The solutions x; y; ´/ of a single linear equation ax C by C c´ D d form a plane in R3 when a, b , and c are not all zero Construct sets of three linear equations whose graphs (a) intersect in a single line, (b) intersect in a single point, and (c) have c Define an appropriate linear transformation T using the matrix in (b), and restate the problem in terms of T 31 Consider the problem of determining whether the following system of equations is consistent for all b1 , b2 , b3 : 2x1 4x2 2x3 D b1 5x1 C x2 C x3 D b2 7x1 5x2 3x3 D b3 a Define appropriate vectors, and restate the problem in terms of Span fv1 ; v2 ; v3 g Then solve that problem Chapter b Define an appropriate matrix, and restate the problem using the phrase “columns of A.” c Define an appropriate linear transformation T using the matrix in (b), and restate the problem in terms of T 32 Describe the possible echelon forms of the matrix A Use the notation of Example in Section 1.2 a A is a matrix whose columns span R2 matrix whose columns span R3 Ä 33 Write the vector as the sum of two vectors, one on the line f.x; y/ W y D 2xg and one on the line f.x; y/ W y D x=2g b A is a 34 Let a1 ; a2 , and b be the vectors in R2 shown in the figure, and let A D Œa1 a2 Does the equation Ax D b have a solution? If so, is the solution unique? Explain x2 b a1 x1 a2 35 Construct a matrix A, not in echelon form, such that the solution of Ax D is a line in R3 Supplementary Exercises 41 Explain why a set fv1 ; v2 ; v3 ; v4 g in R5 must be linearly independent when fv1 ; v2 ; v3 g is linearly independent and v4 is not in Span fv1 ; v2 ; v3 g 42 Suppose fv1 ; v2 g is a linearly independent set in Rn Show that fv1 C v2 ; v1 v2 g is also linearly independent 43 Suppose v1 ; v2 ; v3 are distinct points on one line in R3 The line need not pass through the origin Show that fv1 ; v2 ; v3 g is linearly dependent 44 Let T W Rn ! Rm be a linear transformation, and suppose T u/ D v Show that T u/ D v 45 Let T W R3 ! R3 be the linear transformation that reflects each vector through the plane x2 D That is, T x1 ; x2 ; x3 / D x1 ; x2 ; x3 / Find the standard matrix of T 46 Let A be a 3 matrix with the property that the linear transformation x 7! Ax maps R3 onto R3 Explain why the transformation must be one-to-one 47 A Givens rotation is a linear transformation from Rn to Rn used in computer programs to create a zero entry in a vector (usually a column of a matrix) The standard matrix of a Givens rotation in R2 has the form Ä a b ; a2 C b D b a Ä Ä 10 26 Find a and b such that is rotated into 24 x2 36 Construct a matrix A, not in echelon form, such that the solution of Ax D is a plane in R3 (10, 24) 37 Write the reduced echelon form of a 3 matrix A such that the 3first 2two3 columns of A are pivot columns and A4 D 38 Determine the value(s) of a such that linearly independent Ä Ä aC2 ; a aC6 (26, 0) 40 Use Theorem in Section 1.7 to explain why the columns of the matrix A are linearly independent 0 62 07 AD6 43 05 10 x1 A Givens rotation in R2 is 39 In (a) and (b), suppose the vectors are linearly independent What can you say about the numbers a; : : : ; f ? Justify your answers [Hint: Use a theorem for (b).] 3 3 3 a b d a b d 617 6c7 e 7 7 a 5, c 5, e b 5, 5, f 0 f 0 119 48 The following equation describes a Givens rotation in R3 Find a and b 40 0 a b 32 3 2 b 54 D 5 ; a a2 C b D 49 A large apartment building is to be built using modular construction techniques The arrangement of apartments on any particular floor is to be chosen from one of three basic floor plans Plan A has 18 apartments on one floor, including three-bedroom units, two-bedroom units, and one-bedroom units Each floor of plan B includes threebedroom units, two-bedroom units, and one-bedroom units Each floor of plan C includes three-bedroom units, 120 CHAPTER Linear Equations in Linear Algebra two-bedroom units, and one-bedroom units Suppose the building contains a total of x1 floors of plan A, x2 floors of plan B, and x3 floors of plan C 3 a What interpretation can be given to the vector x1 5? b Write a formal linear combination of vectors that expresses the total numbers of three-, two-, and onebedroom apartments contained in the building T c Is it possible to design the building with exactly 66 three-bedroom units, 74 two-bedroom units, and 136 onebedroom units? If so, is there more than one way to it? Explain your answer ... CHAPTER Linear Equations in Linear Algebra rocks and are measured by geophones attached to mile-long cables behind the ship relies on linear algebra techniques and systems of linear equations Linear. .. asserted by them in accordance with the Copyright, Designs and Patents Act 1988 Authorized adaptation from the United States edition, entitled Linear Algebra and Its Applications, 6th Edition, ... of students and teachers to the first five editions of Linear Algebra and Its Applications has been most gratifying This Sixth Edition provides substantial support both for teaching and for using