A Bridge to Linear Algebra

507 Pages·2019·5.913 MB·English

by  Dragu Atanasiu, Piotr Mikusiński

A BRIDGE TO
LINEAR
ALGEBRA
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A BRIDGE TO
LINEAR
ALGEBRA
Dragu Atanasiu
University of Borås, Sweden
Piotr Mikusiński
University of Central Florida, USA
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO
11276_9789811200229_TP.indd 2 15/2/19 4:38 PM
Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data
Names: Atanasiu, Dragu, author. | Mikusiński, Piotr, author.
Title: A bridge to linear algebra / by Dragu Atanasiu (University of Borås, Sweden),
Piotr Mikusiński (University of Central Florida, USA).
Description: New Jersey : World Scientific, 2019. | Includes index.
Identifiers: LCCN 2018061427| ISBN 9789811200229 (hardcover : alk. paper) |
ISBN 9789811201462 (pbk. : alk. paper)
Subjects: LCSH: Algebras, Linear--Textbooks. | Algebra--Textbooks.
Classification: LCC QA184.2 .A83 2019 | DDC 512/.5--dc23
LC record available at https://lccn.loc.gov/2018061427
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2019 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or
mechanical, including photocopying, recording or any information storage and retrieval system now known or to
be invented, without written permission from the publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center,
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the publisher.
For any available supplementary material, please visit
https://www.worldscientific.com/worldscibooks/10.1142/11276#t=suppl
Printed in Singapore
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Wededicatethisbooktoourwives,
DeliaandGraz˙yna
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Contents
Preface ix
1 Basicideasoflinearalgebra 1
1.1 2×2matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Inversematrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.4 Diagonalizationof2×2matrices . . . . . . . . . . . . . . . . . . . . . . . 39
2 Matrices 55
2.1 Generalmatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.2 Gaussianelimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.3 Theinverseofamatrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3 Thevectorspace(cid:82)2 131
3.1 Vectorsin(cid:82)2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
3.2 Thedotproductandtheprojectiononavectorlinein(cid:82)2 . . . . . . . . 143
3.3 Symmetric2×2matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
4 Thevectorspace(cid:82)3 179
4.1 Vectorsin(cid:82)3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
4.2 Projectionsin(cid:82)3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
5 Determinantsandbasesin(cid:82)3 233
5.1 Thecrossproduct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
5.2 Calculatinginversesanddeterminantsof3×3matrices . . . . . . . . . 250
5.3 Lineardependenceofthreevectorsin(cid:82)3 . . . . . . . . . . . . . . . . . . 264
5.4 Thedimensionofavectorsubspaceof(cid:82)3. . . . . . . . . . . . . . . . . . 283
6 Singularvaluedecompositionof3×2matrices 291
7 Diagonalizationof 3×3matrices 307
7.1 Eigenvaluesandeigenvectorsof 3×3matrices . . . . . . . . . . . . . . 307
7.2 Symmetric3×3matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
vii
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viii CONTENTS
8 Applicationstogeometry 355
8.1 Linesin(cid:82)2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
8.2 Linesandplanesin(cid:82)3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370
9 Rotations 391
9.1 Rotationsin(cid:82)2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
9.2 Quadraticforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400
9.3 Rotationsin(cid:82)3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414
9.4 Crossproductandtheright-handrule. . . . . . . . . . . . . . . . . . . . 420
10Problemsinplanegeometry 429
10.1 Linesandcircles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
10.2 Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433
10.3 Geometryandtrigonometry . . . . . . . . . . . . . . . . . . . . . . . . . 443
10.4 GeometryproblemsfromtheInternationalMathematicalOlympiads . 446
11Problemsforacomputeralgebrasystem 457
12Answerstoselectedexercises 459
Bibliography 491
Index 493
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Preface
Asteachersintheclassroom,wehavenoticedthatsomehardworkingstudentshave
troublefindingtheirfeetwhentacklinglinearalgebraforthefirsttime.Oneofushas
beenteachingstudentsinSwedenformanyyearsusingamoreaccessiblemethod
whichbecametheeventualfoundationandinspirationforthisbook.
Whydoweneedyetanotherlinearalgebratext?
To provide introductory level mathematics students greater opportunities for
success in both grasping, practicing, and internalizing the foundation tools of
Linear Algebra. We present these tools in concrete examples prior to being pre-
sentedwithhigherlevelcomplexconcepts,propertiesandoperations.
TOSTUDENTS:
This book is intended to be read, with or without help from an instructor, as
anintroductiontothegeneraltheorypresentedinastandardlinearalgebracourse.
Students are encouraged to read it before or parallel with a standard linear alge-
bratextbookasastudyguide,practicebook,orreferencesourceforwhateverand
whenevertheyhaveproblemsunderstandingthegeneraltheory.Thisbookcanalso
be recommended as a student aid and its material assigned by an instructor as a
referencesourceforstudentsneedingsomecoaching,clarification,orPRACTICE!
Itisourgoaltoprovidea“lifesaver”forstudentsdrowninginastandardlinear
algebra course. When students get confused, lost, or stuck with a general result,
theycanfindaparticularcaseofthatresultinthisbookdonewithallthedetailsand
consequentlyeasytoread.Thenthegeneralresultwillmakemuchmoresense.
Wewelcomestudentstousethisguidetobecomemorecomfortable,confident,
andsuccessfulinunderstandingtheconceptsandtoolsoflinearalgebra.
GOODLUCK!
TOINSTRUCTORS:
Let’s face it, many students experience difficulties when they learn linear
algebra for the first time. For example, they struggle to understand concepts like
linearindependenceandbases. Inordertohelpstudentsweproposethefollowing
pedagogical approach: We present in depth all major topics of a standard course
inlinearalgebrainthecontextof(cid:82)2and(cid:82)3,includinglinearindependence,bases,
dimension, change of basis, rank theorem, rank nullity theorem, orthogonality,
ix