Linear Algebra DeFranza Linear Algebra Introduction to Linear Algebra with Applications by Jim DeFranza and Daniel Gagliardi provides the proper balance between computation, problem solving, and abstraction that will equip students with the necessary skills and problem solving strategies to allow for a greater understanding and appreciation of linear algebra and its numerous applications. Introduction to Linear Algebra with Applications provides students with the necessary tools for success: Linear Algebra Abstract theory is essential to understanding how linear algebra is applied. Each concept is fully developed presenting natural connections between topics giving students a working knowledge of the theory and techniques for each module covered. Applications have been carefully chosen to highlight the utility of linear algebra in order to see the relevancy of the subject matter in other areas of science as well as in mathematics.
Ranging from routine to more challenging, each exercise set extends the concepts or techniques by asking the student to construct complete arguments. End of chapter True/False MD DALIM 976667 7/29/08 CYAN MAG YELO BLACK questions help students connect concepts and facts presented in the chapter. Examples are designed to develop intuition and prepare students to think more conceptually about new topics as they are introduced. Students are introduced to the study of linear algebra in a sequential and thorough manner through an engaging writing style gaining a clear understanding of the theory essential for applying linear algebra to mathematics or other fields of science.
Summaries conclude each section with important facts and techniques providing students with easy access to the material needed to master the exercise sets.com/defranza ISBN 978-0-07-353235-6 MHID 0-07-353235-5 www.com Revised Pages www.com INTRODUCTION TO LINEAR ALGEBRA WITH APPLICATIONS Jim DeFranza St. Lawrence University Dan Gagliardi SUNY Canton First Pages www.com 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.
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Leick Compositor: Laserwords Private Limited Typeface: 10. Donnelly Crawfordsville, IN Library of Congress Cataloging-in-Publication Data DeFranza, James, 1950– Introduction to linear algebra / James DeFranza, Daniel Gagliardi. ISBN 978–0–07–353235–6—ISBN 0–07–353235–5 (hard copy : alk. Algebras, Linear—Textbooks.
Algebras, Linear—Problems, exercises, etc.com Revised Pages www.com To Regan, Sara, and David —JD To Robin, Zachary, Michael, and Eric —DG Revised Pages www.com Revised Confirming Pages www.com 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. 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. in Pure Mathematics from Kent State University in 1979. DeFranza has coauthored PRECALCULUS, Fourth Edition and two other texts in single variable and multivariable calculus.
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.
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. in Pure Mathematics from North Carolina State University in 2003 under the supervision of Aloysius Helminck. 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. 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. Gagliardi lives in northern New York in the picturesque Saint Lawrence River Valley with his wife Robin, and children Zachary, Michael, and Eric. v Revised Pages www.com Contents Preface ix CHAPTER 1 Systems of Linear Equations and Matrices 1 1.1 Systems of Linear Equations 2 Exercise Set 1.2 Matrices and Elementary Row Operations 14 Exercise Set 1.3 Matrix Algebra 26 Exercise Set 1.4 The Inverse of a Square Matrix 39 Exercise Set 1.5 Matrix Equations 48 Exercise Set 1.6 Determinants 54 Exercise Set 1.7 Elementary Matrices and LU Factorization 68 Exercise Set 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 ⺢n 94 Exercise Set 2.2 Linear Combinations 101 Exercise Set 2.3 Linear Independence 111 Exercise Set 2.3 120 Review Exercises 123 Chapter Test 125 vi Revised Pages www.com Contents vii CHAPTER 3 Vector Spaces 127 3.1 Definition of a Vector Space 129 Exercise Set 3.2 Subspaces 140 Exercise Set 3.3 Basis and Dimension 156 Exercise Set 3.4 Coordinates and Change of Basis 173 Exercise Set 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 4.2 The Null Space and Range 214 Exercise Set 4.3 Isomorphisms 226 Exercise Set 4.4 Matrix Representation of a Linear Transformation 235 Exercise Set 4.5 Similarity 249 Exercise Set 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 5.2 Diagonalization 287 Exercise Set 5.3 Application: Systems of Linear Differential Equations 300 Exercise Set 5.3 309 Revised Pages www.com 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 ⺢n 323 Exercise Set 6.2 Inner Product Spaces 333 Exercise Set 6.3 Orthonormal Bases 342 Exercise Set 6.4 Orthogonal Complements 355 Exercise Set 6.5 Application: Least Squares Approximation 366 Exercise Set 6.6 Diagonalization of Symmetric Matrices 377 Exercise Set 6.7 Application: Quadratic Forms 385 Exercise Set 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 Revised Pages www.com 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. 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.
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.
This choice also enables us to use determinants as a theoretical tool throughout the text whenever the need arises. ix Revised Pages www. 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.
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. Euclidean spaces ⺢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. We have found that this approach works well for students with limited exposure to abstraction at this level.
Geometric representations: Whenever possible, we include figures with geomet- ric representations and interpretations to illuminate the ideas being presented. 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.
True/false chapter tests: Each chapter ends with a true/false Chapter Test with approximately 40 questions.