MATH 218 Other Textbook, Fall 2012

Linear Algebra INTRODUCTION TO LINEAR ALGEBRA WITH APPLICATIONS Jim DeFranza St. Lawrence University Dan Gagliardi SUNY Canton INTRODUCTION TO LINEAR ALGEBRA WITH APPLICATIONS Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, New York, NY 10020. Copyright  2009 by The McGraw-Hill Companies, Inc. All rights reserved. No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written consent of The McGraw-Hill Companies, Inc., including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning. Some ancillaries, including electronic and print components, may not be available to customers outside the United States. This book is printed on acid-free paper. 1 2 3 4 5 6 7 8 9 0 DOC/DOC 0 9 8 ISBN 978–0–07–353235–6 MHID 0–07–353235–5 Editorial Director: Stewart K. Mattson Senior Sponsoring Editor: Elizabeth Covello Director of Development: Kristine Tibbetts Developmental Editor: Michelle Driscoll Director of Marketing: Ryan Blankenship Marketing Coordinator: Sabina Navsariwala Project Manager: Joyce Watters Senior Production Supervisor: Sherry L. Kane Senior Media Project Manager: Tammy Juran Designer: Laurie B. Janssen Cover Designer: Ron Bissell (USE) Cover Image: Royalty-Free/CORBIS Senior Photo Research Coordinator: John C. Leland Supplement Producer: Melissa M. Leick Compositor: Laserwords Private Limited Typeface: 10.25/12 Times Printer: R. R. Donnelly Crawfordsville, IN Library of Congress Cataloging-in-Publication Data DeFranza, James, 1950– Introduction to linear algebra / James DeFranza, Daniel Gagliardi. —1st ed. p. cm. Includes index. ISBN 978–0–07–353235–6—ISBN 0–07–353235–5 (hard copy : alk. paper) 1. Algebras, Linear—Textbooks. 2. Algebras, Linear—Problems, exercises, etc. I. Gagliardi, Daniel. II. Title. QA184.2.D44 2009 515′.5—dc22 2008026020 www.mhhe.com To Regan, Sara, and David —JD To Robin, Zachary, Michael, and Eric —DG About the Authors Jim DeFranza was born in 1950 in Yonkers New York and grew up in Dobbs Ferry New York on the Hudson River. Jim DeFranza is Professor of Mathematics at St. Lawrence University in Canton New York where he has taught undergraduate mathematics for 25 years. St. Lawrence University is a small Liberal Arts College in upstate New York that prides itself in the close interaction that exists between students and faculty. It is this many years of working closely with students that has shaped this text in Linear Algebra and the other texts he has written. He received his Ph.D. in Pure Mathematics from Kent State University in 1979. Dr. DeFranza has coauthored PRECALCULUS, Fourth Edition and two other texts in single variable and multivariable calculus. Dr. DeFranza has also published a dozen research articles in the areas of Sequence Spaces and Classical Summability Theory. Jim is married and has two children David and Sara. Jim and his wife Regan live outside of Canton New York in a 150 year old farm house. Daniel Gagliardi is an Assistant Professor of Mathematics at SUNY Canton, in Canton New York. Dr. Gagliardi began his career as a software engineer at IBM in East Fishkill New York writing programs to support semiconductor development and manufacturing. He received his Ph.D. in Pure Mathematics from North Carolina State University in 2003 under the supervision of Aloysius Helminck. Dr. Gagliardi’s principle area of research is in Symmetric Spaces. In particular, his current work is concerned with developing algorithmic formulations to describe the fine structure (characters and Weyl groups) of local symmetric spaces. Dr. Gagliardi also does research in Graph Theory. His focus there is on the graphical realization of certain types of sequences. In addition to his work as a mathematician, Dr. Gagliardi is an accomplished double bassist and has recently recorded a CD of jazz standards with Author/Pianist Bill Vitek. Dr. Gagliardi lives in northern New York in the picturesque Saint Lawrence River Valley with his wife Robin, and children Zachary, Michael, and Eric. v Contents Preface ix CHAPTER 1 Systems of Linear Equations and Matrices 1 1.1 Systems of Linear Equations 2 Exercise Set 1.1 12 1.2 Matrices and Elementary Row Operations 14 Exercise Set 1.2 23 1.3 Matrix Algebra 26 Exercise Set 1.3 37 1.4 The Inverse of a Square Matrix 39 Exercise Set 1.4 45 1.5 Matrix Equations 48 Exercise Set 1.5 51 1.6 Determinants 54 Exercise Set 1.6 65 1.7 Elementary Matrices and LU Factorization 68 Exercise Set 1.7 77 1.8 Applications of Systems of Linear Equations 79 Exercise Set 1.8 84 Review Exercises 89 Chapter Test 90 CHAPTER 2 Linear Combinations and Linear Independence 93 2.1 Vectors in �n 94 Exercise Set 2.1 99 2.2 Linear Combinations 101 Exercise Set 2.2 108 2.3 Linear Independence 111 Exercise Set 2.3 120 Review Exercises 123 Chapter Test 125 vi Contents vii CHAPTER 3 Vector Spaces 127 3.1 Definition of a Vector Space 129 Exercise Set 3.1 137 3.2 Subspaces 140 Exercise Set 3.2 154 3.3 Basis and Dimension 156 Exercise Set 3.3 171 3.4 Coordinates and Change of Basis 173 Exercise Set 3.4 182 3.5 Application: Differential Equations 185 Exercise Set 3.5 193 Review Exercises 194 Chapter Test 195 CHAPTER 4 Linear Transformations 199 4.1 Linear Transformations 200 Exercise Set 4.1 211 4.2 The Null Space and Range 214 Exercise Set 4.2 223 4.3 Isomorphisms 226 Exercise Set 4.3 233 4.4 Matrix Representation of a Linear Transformation 235 Exercise Set 4.4 245 4.5 Similarity 249 Exercise Set 4.5 253 4.6 Application: Computer Graphics 255 Exercise Set 4.6 268 Review Exercises 270 Chapter Test 272 CHAPTER 5 Eigenvalues and Eigenvectors 275 5.1 Eigenvalues and Eigenvectors 276 Exercise Set 5.1 285 5.2 Diagonalization 287 Exercise Set 5.2 298 5.3 Application: Systems of Linear Differential Equations 300 Exercise Set 5.3 309 viii Contents 5.4 Application: Markov Chains 310 Exercise Set 5.4 315 Review Exercises 316 Chapter Test 318 CHAPTER 6 Inner Product Spaces 321 6.1 The Dot Product on �n 323 Exercise Set 6.1 331 6.2 Inner Product Spaces 333 Exercise Set 6.2 341 6.3 Orthonormal Bases 342 Exercise Set 6.3 352 6.4 Orthogonal Complements 355 Exercise Set 6.4 364 6.5 Application: Least Squares Approximation 366 Exercise Set 6.5 375 6.6 Diagonalization of Symmetric Matrices 377 Exercise Set 6.6 383 6.7 Application: Quadratic Forms 385 Exercise Set 6.7 392 6.8 Application: Singular Value Decomposition 392 Exercise Set 6.8 403 Review Exercises 404 Chapter Test 406 Appendix 409 Answers to Odd-Numbered Exercises 440 Index 479 Preface Introduction to Linear Algebra with Applications is an introductory text targeted to second-year or advanced first-year undergraduate students. The organization of this text is motivated by what our experience tells us are the essential concepts that students should master in a one-semester undergraduate linear algebra course. The centerpiece of our philosophy regarding the presentation of the material is that each topic should be fully developed before the reader moves onto the next. In addition, there should be a natural connection between topics. We take great care to meet both of these objec- tives. This allows us to stay on task so that each topic can be covered with the depth required before progression to the next logical one. As a result, the reader is prepared for each new unit, and there is no need to repeat a concept in a subsequent chapter when it is utilized. Linear algebra is taken early in an undergraduate curriculum and yet offers the opportunity to introduce the importance of abstraction, not only in mathematics, but in many other areas where linear algebra is used. Our approach is to take advantage of this opportunity by presenting abstract vector spaces as early as possible. Throughout the text, we are mindful of the difficulties that students at this level have with abstraction and introduce new concepts first through examples which gently illustrate the idea. To motivate the definition of an abstract vector space, and the subtle concept of linear independence, we use addition and scalar multiplication of vectors in Euclidean space. We have strived to create a balance among computation, problem solving, and abstraction. This approach equips students with the necessary skills and problem- solving strategies in an abstract setting that allows for a greater understanding and appreciation for the numerous applications of the subject. Pedagogical Features 1. Linear systems, matrix algebra, and determinants: We have given a stream- lined, but complete, discussion of solving linear systems, matrix algebra, determi- nants, and their connection in Chap. 1. Computational techniques are introduced, and a number of theorems are proved. In this way, students can hone their problem-solving skills while beginning to develop a conceptual sense of the fun- damental ideas of linear algebra. Determinants are no longer central in linear algebra, and we believe that in a course at this level, only a few lectures should be devoted to the topic. For this reason we have presented all the essentials on determinants, including their connection to linear systems and matrix inverses, in Chap. 1. This choice also enables us to use determinants as a theoretical tool throughout the text whenever the need arises. ix x Preface 2. Vectors: Vectors are introduced in Chap. 1, providing students with a familiar structure to work with as they start to explore the properties which are used later to characterize abstract vector spaces. 3. Linear independence: We have found that many students have difficulties with linear combinations and the concept of linear independence. These ideas are fun- damental to linear algebra and are essential to almost every topic after linear systems. When students fail to grasp them, the full benefits of the course cannot be realized. In Introduction to Linear Algebra with Applications we have devoted Chap. 2 to a careful exposition of linear combinations and linear independence in the context of Euclidean space. This serves several purposes. First, by placing these concepts in a separate chapter their importance in linear algebra is high- lighted. Second, an instructor using the text can give exclusive focus to these ideas before applying them to other problems and situations. Third, many of the impor- tant ramifications of linear combinations and linear independence are considered in the familiar territory of Euclidean spaces. 4. Euclidean spaces �n: The Euclidean spaces and their algebraic properties are introduced in Chap. 2 and are used as a model for the abstract vectors spaces of Chap. 3. We have found that this approach works well for students with limited exposure to abstraction at this level. 5. Geometric representations: Whenever possible, we include figures with geomet- ric representations and interpretations to illuminate the ideas being presented. 6. New concepts: New concepts are almost always introduced first through concrete examples. Formal definitions and theorems are then given to describe the situation in general. Additional examples are also provided to further develop the new idea and to explore it in greater depth. 7. True/false chapter tests: Each chapter ends with a true/false Chapter Test with approximately 40 questions. These questions are designed to help the student connect concepts and better understand the facts presented in the chapter. 8. Rigor and intuition: The approach we have taken attempts to strike a balance between presenting a rigorous development of linear algebra and building intu- ition. For example, we have chosen to omit the proofs for theorems that are not especially enlightening or that contain excessive computations. When a proof is not present, we include a motivating discussion describing the importance and use of the result and, if possible, the idea behind a proof. 9. Abstract vector spaces: We have positioned abstract vector spaces as a central topic within Introduction to Linear Algebra with Applications by placing their introduction as early as possible in Chap. 3. We do this to ensure that abstract vector spaces receive the appropriate emphasis. In a typical undergraduate math- ematics curriculum, a course on linear algebra is the first time that students are exposed to this level of abstraction. However, Euclidean spaces still play a central role in our approach because of their familiarity and since they are so widely used. At the end of this chapter, we include a section on differential equations which underscores the need for the abstract theory of vector spaces. Preface xi 10. Section fact summaries: Each section ends with a summary of the important facts and techniques established in the section. They are written, whenever possible, using nontechnical language and mostly without notation. These summaries are not meant to give a recapitulation of the details and formulas of the section; rather they are designed to give an overview of the main ideas of the section. Our intention is to help students to make connections between the concepts of the section as they survey the topic from a greater vantage point. Applications Over the last few decades the applications of linear algebra have mushroomed, increas- ing not only in their numbers, but also in the diversity of fields to which they apply. Much of this growth is fueled by the power of modern computers and the availability of computer algebra systems used to carry out computations for problems involving large matrices. This impressive power has made linear algebra more relevant than ever. Recently, a consortium of mathematics educators has placed its importance, rel- ative to applications, second only to calculus. Increasingly, universities are offering courses in linear algebra that are specifically geared toward its applications. Whether the intended audience is engineering, economics, science, or mathematics students, the abstract theory is essential to understanding how linear algebra is applied. In this text our introduction to the applications of linear algebra begins in Sec. 1.8 where we show how linear systems can be used to solve problems related to chemistry, engineering, economics, nutrition, and urban planning. However, many types of appli- cations involve the more sophisticated concepts we develop in the text. These appli- cations require the theoretical notions beyond the basic ideas of Chap. 1, and are presented at the end of a chapter as soon as the required background material is com- pleted. Naturally, we have had to limit the number of applications considered. It is our hope that the topics we have chosen will interest the reader and lead to further inquiry. Specifically, in Sec. 4.6, we discuss the role of linear algebra in computer graph- ics. An introduction to the connection between differential equations and linear algebra is given in Secs. 3.5 and 5.3. Markov chains and quadratic forms are examined in Secs. 5.4 and 6.7, respectively. Section 6.5 focuses on the problem of finding approx- imate solutions to inconsistent linear systems. One of the most familiar applications here is the problem of finding the equation of a line that best fits a set of data points. Finally, in Sec. 6.8 we consider the singular value decomposition of a matrix and its application to data compression. Technology Computations are an integral part of any introductory course in mathematics and certainly in linear algebra. To gain mastery of the techniques, we encourage the student to solve as many problems as possible by hand. That said, we also encourage the student to make appropriate use of the available technologies designed to facilitate, or to completely carry out, some of the more tedious computations. For example, it is quite reasonable to use a computer algebra system, such as MAPLE or MATLAB, xii Preface to row-reduce a large matrix. Our approach in Introduction to Linear Algebra with Applications is to assume that some form of technology will be used, but leave the choice to the individual instructor and student. We do not think that it is necessary to include discussions or exercises that use particular software. Note that this text can be used with or without technology. The degree to which it is used is left to the discretion of the instructor. From our own experience, we have found that Scientific Notebook,TM which offers a front end for LATEX along with menu access to the computer algebra system MuPad, allows the student to gain experience using technology to carry out computations while learning to write clear mathematics. Another option is to use LATEX for writing mathematics and a computer algebra system to perform computations. Another aspect of technology in linear algebra has to do with the accuracy and efficiency of computations. Some applications, such as those related to Internet search engines, involve very large matrices which require extensive processing. Moreover, the accuracy of the results can be affected by computer roundoff error. For example, using the characteristic equation to find the eigenvalues of a large matrix is not feasible. Overcoming problems of this kind is extremely important. The field of study known as numerical linear algebra is an area of vibrant research for both software engineers and applied mathematicians who are concerned with developing practical solutions. In our text, the fundamental concepts of linear algebra are introduced using simple examples. However, students should be made aware of the computational difficulties that arise when extending these ideas beyond the small matrices used in the illustrations. Other Features 1. Chapter openers: The opening remarks for each chapter describe an application that is directly related to the material in the chapter. These provide additional motivation and emphasize the relevance of the material that is about to be covered. 2. Writing style: The writing style is clear, engaging, and easy to follow. Impor- tant new concepts are first introduced with examples to help develop the reader’s intuition. We limit the use of jargon and provide explanations that are as reader- friendly as possible. Every explanation is crafted with the student in mind. Intro- duction to Linear Algebra with Applications is specifically designed to be a readable text from which a student can learn the fundamental concepts in linear algebra. 3. Exercise sets: Exercise sets are organized with routine exercises at the beginning and the more difficult problems toward the end. There is a mix of computational and theoretical exercises with some requiring proof. The early portion of each exercise set tests the student’s ability to apply the basic concepts. These exercises are primarily computational, and their solutions follow from the worked examples in the section. The latter portion of each exercise set extends the concepts and techniques by asking the student to construct complete arguments. 4. Review exercise sets: The review exercise sets are organized as sample exams with 10 exercises. These exercises tend to have multiple parts, which connect the various techniques and concepts presented in the text. At least one problem in each of these sets presents a new idea in the context of the material of the chapter. Preface xiii 5. Length: The length of the text reflects the fact that it is specifically designed for a one-semester course in linear algebra at the undergraduate level. 6. Appendix: The appendix contains background material on the algebra of sets, functions, techniques of proof, and mathematical induction. With this feature, the instructor is able to cover, as needed, topics that are typically included in a Bridge Course to higher mathematics. Course Outline The topics we have chosen for Introduction to Linear Algebra with Applications closely follow those commonly covered in a first introductory course. The order in which we present these topics reflects our approach and preferences for emphasis. Nevertheless, we have written the text to be flexible, allowing for some permutations of the order of topics without any loss of consistency. In Chap. 1 we present all the basic material on linear systems, matrix algebra, determinants, elementary matrices, and the LU decomposition. Chap. 2 is entirely devoted to a careful exposition of lin- ear combinations and linear independence in �n. We have found that many students have difficulty with these essential concepts. The addition of this chapter gives us the opportunity to develop all the important ideas in a familiar setting. As mentioned earlier, to emphasize the importance of abstract vector spaces, we have positioned their introduction as early as possible in Chap. 3. Also, in Chap. 3 is a discussion of subspaces, bases, and coordinates. Linear transformations between vector spaces are the subject of Chap. 4. We give descriptions of the null space and range of a linear transformation at the beginning of the chapter, and later we show that every finite dimensional vector space, of dimension n, is isomorphic to �n. Also, in Chap. 4 we introduce the four fundamental subspaces of a matrix and discuss the action of an m × n matrix on a vector in �n. Chap. 5 is concerned with eigenvalues and eigenvec- tors. An abundance of examples are given to illustrate the techniques of computing eigenvalues and finding the corresponding eigenvectors. We discuss the algebraic and geometric multiplicities of eigenvalues and give criteria for when a square matrix is diagonalizable. In Chap. 6, using �n as a model, we show how a geometry can be defined on a vector space by means of an inner product. We also give a description of the Gram-Schmidt process used to find an orthonormal basis for an inner product space and present material on orthogonal complements. At the end of this chapter we discuss the singular value decomposition of an m × n matrix. The Appendix contains a brief summary of some topics found in a Bridge Course to higher mathematics. Here we include material on the algebra of sets, functions, techniques of proof, and mathematical induction. Application sections are placed at the end of chapters as soon as the requisite background material has been covered. Supplements 1. Instructor solutions manual: This manual contains detailed solutions to all exercises. 2. Student solutions manual: This manual contains detailed solutions to odd- numbered exercises. xiv Preface 3. Text website www.mhhe.com/defranza: This website accompanies the text and is available for both students and their instructors. Students will be able to access self-assessment quizzes and extra examples for each section and end of chapter cumulative quizzes. In addition to these assets, instructors will be able to access additional quizzes, sample exams, the end of chapter true/false tests, and the Instructor’s Solutions Manual. Acknowledgments We would like to give our heartfelt thanks to the many individuals who reviewed the manuscript at various stages of its development. Their thoughtful comments and excellent suggestions have helped us enormously with our efforts to realize our vision of a reader-friendly introductory text on linear algebra. We would also like to give special thanks to David Meel of Bowling Green State University, Bowling Green, Ohio, for his thorough review of the manuscript and insightful comments that have improved the exposition of the material in the text. We are also grateful to Ernie Stitzinger of North Carolina State University who had the tiring task of checking the complete manuscript for accuracy, including all the exercises. A very special thanks goes to our editors (and facilitators), Liz Covello (Sr. Sponsoring Editor), Michelle Driscoll (Developmental Editor), and Joyce Watters (Project Manager) who have helped us in more ways than we can name, from the inception of this project to its completion. On a personal level, we would like to thank our wives, Regan DeFranza and Robin Gagliardi, for their love and support; and our students at Saint Lawrence University and SUNY Canton who provided the motivation to write the text. Finally, we want to express our gratitude to the staff at McGraw-Hill Higher Education, Inc., for their work in taking our manuscript and producing the text. Preface xv . List of Reviewers Marie Aratari, Oakland Community College Cik Azizah, Universiti Utara Malaysia (UUM) Przcmyslaw Bogacki, Old Dominion University Rita Chattopadhyay, Eastern Michigan University Eugene Don, Queens College Lou Giannini, Curtin University of Technology Gregory Gibson, North Carolina A&T University Mark Gockenback, Michigan Technological University Dr. Leong Wah June, Universiti Putra Malaysia Cerry Klein, University of Missouri–Columbia Kevin Knudson, Mississippi State University Hyungiun Ko, Yonsei University Jacob Kogan, University of Maryland–Baltimore County David Meel, Bowling Green State University Martin Nakashima, California State Poly University–Pomona Eugene Spiegel, University of Connecticut–Storrs Dr. Hajar Sulaiman, Universiti Sains Malaysia (USM) Gnana Bhaskar Tenali, Florida Institute of Technology–Melbourne Peter Wolfe, University of Maryland–College Park To The Student You are probably taking this course early in your undergraduate studies after two or three semesters of calculus, and most likely in your second year. Like calculus, linear algebra is a subject with elegant theory and many diverse applications. However, in this course you will be exposed to abstraction at a much higher level. To help with this transition, some colleges and universities offer a Bridge Course to Higher Mathematics. If you have not already taken such a course, this may likely be the first mathematics course where you will be expected to read and understand proofs of theorems, provide proofs of results as part of the exercise sets, and apply the concepts presented. All this is in the context of a specific body of knowledge. If you approach this task with an open mind and a willingness to read the text, some parts perhaps more than once, it will be an exciting and rewarding experience. Whether you are taking this course as part of a mathematics major or because linear algebra is applied in your specific area of study, a clear understanding of the theory is essential for applying the concepts of linear algebra to mathematics or other fields of science. The solved examples and exercises in the text are designed to prepare you for the types of problems you can expect to see in this course and other more advanced courses in mathematics. The organization of the material is based on our philosophy that each topic should be fully developed before readers move onto the next. The image of a tree on the front cover of the text is a metaphor for this learning strategy. It is particularly applicable to the study of mathematics. The trunk of the tree represents the material that forms the basis for everything that comes afterward. In our text, this material is contained in Chaps. 1 through 4. All other branches of the tree, representing more advanced topics and applications, extend from the foundational material of the trunk or from the ancillary material of the intervening branches. We have specifically designed our text so that you can read it and learn the concepts of linear algebra in a sequential and thorough manner. If you remain committed to learning this beautiful subject, the rewards will be significant in other courses you may take, and in your professional career. Good luck! Jim DeFranza jdefranza@stlawu.edu Dan Gagliardi gagliardid@canton.edu xvi Applications Index Aircraft Design, 199 Astronomy, 61 Average Global Temperatures, 371 Balancing Chemical Equations, 1, 79, 84 Carbon Emissions, 321 Cellular Phone Subscribers, 86 Computer Graphics, 199, 255 Cumulative HIV Infections, 376 Data Compression, 401 Differential Equations, 185 Digital Transmissions, 127, 402 Economics, 82, 86, 87 Electrical Networks, 88, 191 Engineering, 185, 191 Epidemics, 87, 315 Exponential Growth and Decay, 186 Fitting Data with a Curve, 11, 61, 366, 372, 376 Fourier Approximation, 373 Infant Mortality Rates, 376 Least Squares Approximation, 321, 366 Linear Regression, 371 Markov Chains, 275, 310 Mass Transit Systems, 315 Mixing, 307, 310 National Health Care, 86 Networks, 79, 85, 89 Nutrition, 81, 85 Photosynthesis, 1 Plant Breading, 315 Population Distribution, 87, 275, 315 Powers of a Matrix, 287 Predator–Prey Models, 300 Projections, 265, 343, 345, 361, 365 Quadratic Forms, 385 Rotation of Axes, 385 Search Engines, 276 xvii xviii Applications Index Signal Processing, 93 Singular Value Decomposition, 392 Systems of Differential Equations, 300 Thermal Equilibrium, 88, 310 Vibrating Spring, 192, 194 World Hydroelectricity Use, 376 World Population, 376 C H A P T E R Systems of Linear Equations and Matrices CHAPTER OUTLINE 1.1 Systems of Linear Equations 2 1.2 Matrices and Elementary Row Operations 14 1.3 Matrix Algebra 26 1.4 The Inverse of a Square Matrix 39 1.5 Matrix Equations 48 1.6 Determinants 54 1.7 Elementary Matrices and LU Factorization 68 1.8 Applications of Systems of Linear Equations 79 I n the process of photosynthesis solar energy is converted into forms that are used by living organisms. The chemical reaction that occurs in the leaves of plants converts carbon dioxide and water to carbohydrates with the release of oxygen. The chemical equation of the reaction takes the form aCO2 + bH2O → cO2 + dC6H12O6 where a, b, c, and d are some positive whole numbers. The law of conservation of mass states that the total mass of all substances present before and after a chemical reaction remains the same. That is, atoms are neither created nor destroyed Photograph by Jan Smith/RF in a chemical reaction, so chemical equations must be balanced. To balance the pho- tosynthesis reaction equation, the s