Introduction to Linear Algebra with Applications

510 Pages·2015·16.41 MB·English

by  Jim DeFranza

Introduction
to
Linear Algebra
Applications
with

Introduction
to
Linear Algebra
Applications
with
Jim DeFranza
St. Lawrence University
Daniel Gagliardi
SUNY Canton
For information about this book, contact:
Waveland Press, Inc.
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[email protected]
www.waveland.com
Copyright © 2009 by Jim DeFranza and Daniel Gagliardi
Reissued 2015 by Waveland Press, Inc.
10-digit ISBN 1-4786-2777-8
13-digit ISBN 978-1-4786-2777-7
All rights reserved. No part of this book may be reproduced, stored in a retrieval
system, or transmitted in any form or by any means without permission in writing from
the publisher.
Printed in the United States of America
7 6 5 4 3 2 1
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.
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.
vii
Contents
Preface xi
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:31)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
viii
Contents ix
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