Introduction to Linear Algebra

275 Pages·2021·14.761 MB·English

by  Rita Fioresi, Marta Morigi

Introduction to Linear Algebra

Introduction to Linear Algebra
Rita Fioresi
University of Bologna
Marta Morigi
University of Bologna
First edition published 2022
by CRC Press
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Authorized translation from Italian language edition published by CEA – Casa Editrice Ambrosiana, A Division of
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Library of Congress Cataloging‑in‑Publication Data
Names: Fioresi, Rita, 1966- author. | Morigi, Marta, author.
Title: Introduction to linear algebra / Rita Fioresi, University of
Bologna, Marta Morigi, University of Bologna.
Description: First edition. | Boca Raton : Chapman & Hall/CRC Press, 2021.
| Includes bibliographical references and index.
Identifiers: LCCN 2021019430 (print) | LCCN 2021019431 (ebook) | ISBN
9780367626549 (hardback) | ISBN 9780367635503 (paperback) | ISBN
9781003119609 (ebook)
Subjects: LCSH: Algebras, Linear.
Classification: LCC QA184.2 .F56 2021 (print) | LCC QA184.2 (ebook) | DDC
512/.5--dc23
LC record available at https://lccn.loc.gov/2021019430
LC ebook record available at https://lccn.loc.gov/2021019431
ISBN: 978-0-367-62654-9 (hbk)
ISBN: 978-0-367-63550-3 (pbk)
ISBN: 978-1-003-11960-9 (ebk)
Typeset in LM Roman
by KnowledgeWorks Global Ltd.
Contents
Preface ix
Chapter 1(cid:4) Introduction to Linear Systems 1
1.1 LINEARSYSTEMS:FIRSTEXAMPLES 1
1.2 MATRICES 3
1.3 MATRICESANDLINEARSYSTEMS 6
1.4 THEGAUSSIANALGORITHM 11
1.5 EXERCISESWITHSOLUTIONS 16
1.6 SUGGESTEDEXERCISES 22
Chapter 2(cid:4) Vector Spaces 25
2.1 INTRODUCTION:THESETOFREALNUMBERS 25
2.2 THEVECTORSPACERN ANDTHEVECTORSPACEOFMATRICES 26
2.3 VECTORSPACES 31
2.4 SUBSPACES 33
2.5 EXERCISESWITHSOLUTIONS 38
2.6 SUGGESTEDEXERCISES 39
Chapter 3(cid:4) Linear Combination and Linear Independence 41
3.1 LINEARCOMBINATIONSANDGENERATORS 41
3.2 LINEARINDEPENDENCE 47
3.3 EXERCISESWITHSOLUTIONS 51
3.4 SUGGESTEDEXERCISES 54
Chapter 4(cid:4) Basis and Dimension 57
4.1 BASIS:DEFINITIONANDEXAMPLES 57
4.2 THECONCEPTOFDIMENSION 61
4.3 GAUSSIANALGORITHM 64
4.4 EXERCISESWITHSOLUTIONS 68
4.5 SUGGESTEDEXERCISES 72
v
vi (cid:4) Contents
4.6 APPENDIX:THECOMPLETIONTHEOREM 74
Chapter 5(cid:4) Linear Transformations 77
5.1 LINEARTRANSFORMATIONS:DEFINITION 77
5.2 LINEARMAPSANDMATRICES 82
5.3 THECOMPOSITIONOFLINEARTRANSFORMATIONS 84
5.4 KERNELANDIMAGE 86
5.5 THERANKNULLITYTHEOREM 89
5.6 ISOMORPHISMOFVECTORSPACES 91
5.7 CALCULATIONOFKERNELANDIMAGE 92
5.8 EXERCISESWITHSOLUTIONS 95
5.9 SUGGESTEDEXERCISES 97
Chapter 6(cid:4) Linear Systems 101
6.1 PREIMAGE 101
6.2 LINEARSYSTEMS 103
6.3 EXERCISESWITHSOLUTIONS 108
6.4 SUGGESTEDEXERCISES 111
Chapter 7(cid:4) Determinant and Inverse 113
7.1 DEFINITIONOFDETERMINANT 113
7.2 CALCULATINGTHEDETERMINANT:CASES2×2AND3×3 117
7.3 CALCULATINGTHEDETERMINANTWITHARECURSIVE
METHOD 119
7.4 INVERSEOFAMATRIX 121
7.5 CALCULATIONOFTHEINVERSEWITHTHEGAUSSIAN
ALGORITHM 123
7.6 THELINEARMAPSFROMRN TORN 125
7.7 EXERCISESWITHSOLUTIONS 126
7.8 SUGGESTEDEXERCISES 127
7.9 APPENDIX 128
Chapter 8(cid:4) Change of Basis 141
8.1 LINEARTRANSFORMATIONSANDMATRICES 141
8.2 THEIDENTITYMAP 144
8.3 CHANGEOFBASISFORLINEARTRANSFORMATIONS 148
8.4 EXERCISESWITHSOLUTIONS 150
8.5 SUGGESTEDEXERCISES 152
Contents (cid:4) vii
Chapter 9(cid:4) Eigenvalues and Eigenvectors 155
9.1 DIAGONALIZABILITY 155
9.2 EIGENVALUESANDEIGENVECTORS 158
9.3 EXERCISESWITHSOLUTIONS 169
9.4 SUGGESTEDEXERCISES 173
Chapter 10(cid:4) Scalar Products 177
10.1 BILINEARFORMS 177
10.2 BILINEARFORMSANDMATRICES 179
10.3 BASISCHANGE 181
10.4 SCALARPRODUCTS 183
10.5 ORTHOGONALSUBSPACES 185
10.6 GRAM-SCHMIDTALGORITHM 187
10.7 EXERCISESWITHSOLUTIONS 189
10.8 SUGGESTEDEXERCISES 190
Chapter 11(cid:4) Spectral Theorem 193
11.1 ORTHOGONALLINEARTRANSFORMATIONS 193
11.2 ORTHOGONALMATRICES 194
11.3 SYMMETRICLINEARTRANSFORMATIONS 198
11.4 THESPECTRALTHEOREM 199
11.5 EXERCISESWITHSOLUTIONS 202
11.6 SUGGESTEDEXERCISES 203
11.7 APPENDIX:THECOMPLEXCASE 204
Chapter 12(cid:4) Applications of Spectral Theorem and Quadratic Forms 209
12.1 DIAGONALIZATIONOFSCALARPRODUCTS 209
12.2 QUADRATICFORMS 213
12.3 QUADRATICFORMSANDCURVESINTHEPLANE 215
12.4 EXERCISESWITHSOLUTIONS 216
12.5 SUGGESTEDEXERCISES 218
Chapter 13(cid:4) Lines and Planes 221
13.1 POINTSANDVECTORSINR3 221
13.2 SCALARPRODUCTANDVECTORPRODUCT 223
13.3 LINESINR3 226
13.4 PLANESINR3 228
viii (cid:4) Contents
13.5 EXERCISESWITHSOLUTIONS 231
13.6 SUGGESTEDEXERCISES 233
Chapter 14(cid:4) Introduction to Modular Arithmetic 235
14.1 THEPRINCIPLEOFINDUCTION 235
14.2 THEDIVISIONALGORITHMANDEUCLID’SALGORITHM 237
14.3 CONGRUENCECLASSES 241
14.4 CONGRUENCES 244
14.5 EXERCISESWITHSOLUTIONS 246
14.6 SUGGESTEDEXERCISES 247
14.7 APPENDIX:ELEMENTARYNOTIONSOFSETTHEORY 248
Appendix A(cid:4) Complex Numbers 249
A.1 COMPLEXNUMBERS 249
A.2 POLARREPRESENTATION 252
Appendix B(cid:4) Solutions of some suggested exercises 255
Bibliography 261
Index 263
Preface
This textbook comes from the need to cater the essential notions of linear algebra to
Physics,EngineeringandComputerSciencestudents.Westrivedtokeeptheabstrac-
tion and rigor of this beautiful subject and yet give as much as possible the intuition
behind all of the mathematical concepts we introduce. Though we provide the full
proofs of all of our statements, we introduce each topic with a lot of examples and
intuitive explanations to guide the students to a mature understanding.
This is not meant to be a comprehensive treatment on linear algebra but an
essential guide to its foundation and heart for those who want to understand the
basic concepts and the abstract mathematics behind the powerful tools it provides.
A short tour of our presentation goes as follows.
Chapters 1, 13 and 14 are independent from each other and from the rest of the
book. Chapter 1 and/or Chapter 13 can be effectively used as a motivational intro-
duction to linear algebra and vector spaces. Chapter 14 contains some further topics
like the principle of induction and Euclid’s algorithm, which are essential for com-
putersciencestudents,buttheycaneasilybeomittedandthechapterisindependent
from the rest of the book.
Chapters 2, 3, 4 and 5 introduce the basic notions concerning vector spaces and
linear maps, while Chapters 6, 7, 8 and 9 further develop the theory to reach the
question of eigenvalues and eigenvectors. A minimal course in linear algebra can end
after Chapter 6, or even better after Chapter 9. In the remaining Chapters 10, 11
and 12 we study scalar products, the Spectral Theorem and quadratic forms, very
important for physical and engineering applications.
ix