Introduction to Linear Algebra with Applications

509 Pages·2008·5.75 MB·English

by  James DeFranza

DeFranza
Linear Algebra 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:
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Abstract theory is essential to understanding how linear
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algebra is applied.
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Each concept is fully developed presenting natural connections
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between topics giving students a working knowledge of the theory and techniques for each module
covered. a
Applications have been carefully chosen to highlight the utility of linear algebra in r
order to see the relevancy of the subject matter in other areas of science as well as in mathematics.
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Ranging from routine to more challenging, each exercise set extends the concepts
M or techniques by asking the student to construct complete arguments. End of chapter True/False l
D questions help students connect concepts and facts presented in the chapter. g
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A Examples are designed to develop intuition and prepare students to think more e
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9 Students are introduced to the study of linear algebra in a sequential and thorough r
7 manner through an engaging writing style gaining a clear understanding of the theory essential for
666 applying linear algebra to mathematics or other fi elds of science. a
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7 Summaries conclude each section with important facts and techniques providing
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9 students with easy access to the material needed to master the exercise sets.
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RevisedPages
INTRODUCTION TO
LINEAR ALGEBRA
WITH APPLICATIONS
Jim DeFranza
St. Lawrence University
Dan Gagliardi
SUNY Canton
FirstPages
INTRODUCTIONTOLINEARALGEBRAWITHAPPLICATIONS
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LibraryofCongressCataloging-in-PublicationData
DeFranza,James,1950–
Introductiontolinearalgebra/JamesDeFranza,DanielGagliardi.—1sted.
p.cm.
Includesindex.
ISBN978–0–07–353235–6—ISBN0–07–353235–5(hardcopy:alk.paper)
1.Algebras,Linear—Textbooks.2.Algebras,Linear—Problems,exercises,etc.I.Gagliardi,Daniel.II.Title.
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RevisedPages
To Regan, Sara, and David
—JD
To Robin, Zachary, Michael, and Eric
—DG
RevisedPages
RevisedConfirmingPages
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 researcharticles
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.
DanielGagliardi 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 Universityin2003underthe supervisionofAloysiusHelminck.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/PianistBillVitek.Dr.GagliardilivesinnorthernNewYorkinthepicturesque
Saint Lawrence River Valley with his wife Robin, and children Zachary, Michael,
and Eric.
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Contents
Preface ix
1
CHAPTER SystemsofLinearEquationsandMatrices 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
2
CHAPTER LinearCombinationsandLinearIndependence 93
2.1 Vectors in (cid:1)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
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Contents vii
3
CHAPTER VectorSpaces 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
4
CHAPTER LinearTransformations 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
5
CHAPTER EigenvaluesandEigenvectors 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
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5.4 Application: Markov Chains 310
Exercise Set 5.4 315
Review Exercises 316
Chapter Test 318
6
CHAPTER InnerProductSpaces 321
6.1 The Dot Product on (cid:1)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
RevisedPages
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
textismotivatedbywhatourexperiencetellsusaretheessentialconceptsthatstudents
should masterin a one-semesterundergraduate linear algebra course. The centerpiece
of our philosophy regarding the presentation of the material is that eachtopicshould
befullydevelopedbeforethereadermovesontothenext. 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 stayontask so that each topic can be covered with the depth
requiredbefore progressiontothe nextlogicalone.As aresult,the readerisprepared
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
opportunitytointroducetheimportanceofabstraction,notonlyinmathematics,butin
manyotherareaswherelinearalgebraisused.Ourapproachistotakeadvantageofthis
opportunity by presenting abstract vector spaces as early as possible. Throughout the
text,we are mindful of the difficulties thatstudents atthis levelhave withabstraction
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
linearindependence,weuseadditionandscalarmultiplicationofvectorsinEuclidean
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.
PedagogicalFeatures
1. Linear systems, matrix algebra, and determinants: We have given a stream-
lined,butcomplete,discussionofsolvinglinearsystems,matrixalgebra,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.
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