Table Of ContentFundamentals of Linear
Algebra
Fundamentals of Linear
Algebra
J. S. Chahal
Cover photo courtesy of J. S. Chahal, Castle Valley, Utah.
CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2019 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Printed on acid-free paper
Version Date: 20181026
International Standard Book Number-13: 978-1-138-59050-2 (Hardback)
This book contains information obtained from authentic and highly regarded sources. Reasonable
efforts have been made to publish reliable data and information, but the author and publisher cannot
assume responsibility for the validity of all materials or the consequences of their use. The authors and
publishers have attempted to trace the copyright holders of all material reproduced in this publication
and apologize to copyright holders if permission to publish in this form has not been obtained. If any
copyright material has not been acknowledged please write and let us know so we may rectify in any
future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced,
transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or
hereafter invented, including photocopying, microfilming, and recording, or in any information
storage or retrieval system, without written permission from the publishers.
For permission to photocopy or use material electronically from this work, please access
www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc.
(CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization
that provides licenses and registration for a variety of users. For organizations that have been granted
a photocopy license by the CCC, a separate system of payment has been arranged.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and
are used only for identification and explanation without intent to infringe.
Visit the Taylor & Francis Web site at
http://www.taylorandfrancis.com
and the CRC Press Web site at
http://www.crcpress.com
Contents
Preface ix
Advice to the Reader xi
1 Preliminaries 1
1.1 What Is Linear Algebra? . . . . . . . . . . . . . . . . . . . . 1
1.2 Rudimentary Set Theory . . . . . . . . . . . . . . . . . . . . 2
1.3 Cartesian Products . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.5 Concept of a Function . . . . . . . . . . . . . . . . . . . . . . 4
1.6 Composite Functions . . . . . . . . . . . . . . . . . . . . . . 7
1.7 Fields of Scalars . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.8 Techniques for Proving Theorems . . . . . . . . . . . . . . . 11
2 Matrix Algebra 17
2.1 Matrix Operations . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1.1 Addition and Scaling of Matrices:. . . . . . . . . . . . 18
2.1.2 Matrix Multiplication . . . . . . . . . . . . . . . . . . 19
2.2 Geometric Meaning of a Matrix Equation . . . . . . . . . . . 25
2.3 Systems of Linear Equations . . . . . . . . . . . . . . . . . . 27
2.4 Inverse of a Matrix . . . . . . . . . . . . . . . . . . . . . . . 33
2.5 The Equation Ax=b . . . . . . . . . . . . . . . . . . . . . . 38
2.6 Basic Applications . . . . . . . . . . . . . . . . . . . . . . . . 39
2.6.1 Tra(cid:14)c Flow . . . . . . . . . . . . . . . . . . . . . . . . 39
2.6.2 Barter Systems . . . . . . . . . . . . . . . . . . . . . . 40
2.6.3 Electric Circuits . . . . . . . . . . . . . . . . . . . . . 42
2.6.4 Chemical Reactions . . . . . . . . . . . . . . . . . . . 43
2.6.5 Economics. . . . . . . . . . . . . . . . . . . . . . . . . 44
3 Vector Spaces 47
3.1 The Concept of a Vector Space . . . . . . . . . . . . . . . . . 47
3.2 Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.3 The Dimension of a Vector Space . . . . . . . . . . . . . . . 56
3.4 Linear Independence . . . . . . . . . . . . . . . . . . . . . . . 58
3.5 Application of Knowing dim(V) . . . . . . . . . . . . . . . . 63
3.6 Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.7 Rank of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . 68
v
vi Contents
4 Linear Maps 71
4.1 Linear Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2 Properties of Linear Maps . . . . . . . . . . . . . . . . . . . 76
4.3 Matrix of a Linear Map . . . . . . . . . . . . . . . . . . . . . 80
4.4 Matrix Algebra and Algebra of Linear Maps . . . . . . . . . 84
4.5 Linear Functionals and Duality . . . . . . . . . . . . . . . . . 87
4.6 Equivalence and Similarity . . . . . . . . . . . . . . . . . . . 88
4.7 Application to Higher Order Di(cid:11)erential Equations . . . . . 90
5 Determinants 93
5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2 Properties of Determinants . . . . . . . . . . . . . . . . . . . 96
5.3 Existence and Uniqueness of Determinant . . . . . . . . . . . 97
5.4 Computational De(cid:12)nition of Determinant . . . . . . . . . . . 102
5.5 Evaluation of Determinants . . . . . . . . . . . . . . . . . . . 105
5.6 Adjoint and Cramer’s Rule . . . . . . . . . . . . . . . . . . . 108
6 Diagonalization 113
6.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.2 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . 114
6.3 Cayley-Hamilton Theorem . . . . . . . . . . . . . . . . . . . 123
7 Inner Product Spaces 127
7.1 Inner Product . . . . . . . . . . . . . . . . . . . . . . . . . . 127
7.2 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.3 Orthogonal and Orthonormal Sets . . . . . . . . . . . . . . . 134
7.4 Gram-Schmidt Process . . . . . . . . . . . . . . . . . . . . . 136
7.5 Orthogonal Projections on Subspaces . . . . . . . . . . . . . 140
8 Linear Algebra over Complex Numbers 147
8.1 Algebra of Complex Numbers . . . . . . . . . . . . . . . . . 147
8.2 Diagonalization of Matrices with Complex Eigenvalues . . . 150
8.3 Matrices over Complex Numbers . . . . . . . . . . . . . . . . 151
9 Orthonormal Diagonalization 157
9.1 Motivational Introduction . . . . . . . . . . . . . . . . . . . . 157
9.2 Matrix Representation of a Quadratic Form . . . . . . . . . 159
9.3 Spectral Decomposition . . . . . . . . . . . . . . . . . . . . . 161
9.4 Constrained Optimization { Extrema of Spectrum . . . . . . 167
9.5 Singular Value Decomposition (SVD) . . . . . . . . . . . . . 168
10 Selected Applications of Linear Algebra 177
10.1 System of First Order Linear Di(cid:11)erential Equations . . . . . 177
10.2 Multivariable Calculus . . . . . . . . . . . . . . . . . . . . . 179
10.3 Special Theory of Relativity . . . . . . . . . . . . . . . . . . 181
10.4 Cryptography . . . . . . . . . . . . . . . . . . . . . . . . . . 190
Contents vii
10.5 Solving Famous Problems from Greek Geometry . . . . . . . 196
10.5.1 Vector Spaces of Polynomials . . . . . . . . . . . . . . 196
10.5.2 Roots of Polynomials. . . . . . . . . . . . . . . . . . . 198
10.5.3 Straightedge and Compass. . . . . . . . . . . . . . . . 201
10.5.4 Intersecting Lines and Circles . . . . . . . . . . . . . . 203
10.5.5 Degrees of Constructible Numbers . . . . . . . . . . . 204
10.5.6 Solutions of the Famous Problems . . . . . . . . . . . 204
Answers to Selected Numerical Problems 207
Notation 219
Bibliography 221
Index 223
Preface
This book is based on a one-semester course on linear algebra I have taught
numerous times at Brigham Young University. With so many books on the
subject already out there, it is legitimate to ask why one more is necessary.
No textbook on the subject served all my needs while teaching the course.
Forexample,mosttextsdonotprovidesatisfactoryintroductionstosomeim-
portantde(cid:12)nitions.Theirde(cid:12)nitionsseemtoappearsuddenlyoutofnowhere.
So, for my lectures I prepared my own notes, and hence this book. Whenever
possible,Ihavetriedtomotivatebytakingthereaderalongpathswhichlead
naturally to our de(cid:12)nitions. The other salient features of the book are:
1. It gives a brief but adequate presentation of the fundamentals of
the subject in as few pages as possible so that it can be covered in
a semester without missing anything signi(cid:12)cant in textbooks with
500 to 700 pages.
2. Although most students taking the course were non math majors,
therigorhasnotbeencompromised.Tohelpthemcopewithit,the
prerequisite material has been assembled in Chapter 1.
3. Rather than present linear algebra as a hodgepodge collection of
seemingly unrelated topics, I have tried to present it as a single
theme { the study of linear maps { with matrices as a convenient
tooltocaptureandkeeptrackofthembutonlywhentheirdomains
and co-domains are (cid:12)nite dimensional. Conversely, by treating ma-
tricesaslinearmaps,someofthepropertiesofmatricesthemselves,
such as the associativity of matrix multiplication, become obvious.
This reduces the number of pages required for the subject.
4. Contrarytotheusualpracticeofdoinglinearalgebraoverthereals,
or at most over the complex numbers, I have put no restriction on
the (cid:12)eld of scalars. This widens the scope for applications of lin-
ear algebra without increasing the level of di(cid:14)culty or abstraction.
Ironically, when the applications were not the order of the day, the
books on linear algebra of the last generation (e.g., [4], [9], [10] and
[16])usedtobeginwithanintroductionto(cid:12)elds,notonlyasa(cid:12)rst
step towards the abstraction needed for the course but also for its
applications.
ix
