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Elementary Linear Algebra (2016) PDF

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Equivalent Conditions for Singular and Nonsingular Matrices LetAbeann×nmatrix.Anypairofstatementsinthesamecolumnareequivalent. A issingular(A−1 doesnotexist). A isnonsingular(A−1 exists). Rank(A)(cid:2)=n. Rank(A)=n. |A|=0. |A|(cid:2)=0. Aisnot rowequivalenttoIn. AisrowequivalenttoIn. AX=Ohasanontrivial solutionforX. AX=Ohasonlythetrivialsolution forX. AX=Bdoesnot haveaunique AX=Bhasauniquesolution solution(nosolutionsor forX(namely,X=A−1B). infinitelymanysolutions). TherowsofAdonot forma TherowsofAforma basisforRn. basisforRn. ThecolumnsofAdonot form ThecolumnsofAform abasisforRn. abasisforRn. ThelinearoperatorL:Rn→Rn ThelinearoperatorL:Rn→Rn givenbyL(X)=AXis givenbyL(X)=AX not anisomorphism. isanisomorphism. Diagonalization Method Todiagonalize(ifpossible)ann×nmatrixA: Step1: CalculatepA(x)=|xIn−A|. Step2: FindallrealrootsofpA(x)(thatis,allrealsolutionstopA(x) = 0).Thesearethe eigenvaluesλ1,λ2,λ3,...,λkforA. Step3: Foreacheigenvalueλminturn: Row reduce the augmented matrix [λmIn−A|0]. Use the result to obtain a setofparticularsolutionsofthehomogeneoussystem(λmIn−A)X=0bysetting each independentvariableinturnequalto1andallotherindependentvariables equalto0. Step 4: If after repeating Step 3 for each eigenvalue, you have less than n fundamental eigenvectorsoverallforA,thenAcannotbediagonalized.Stop. Step5: Otherwise,formamatrixPwhosecolumnsarethesenfundamentaleigenvectors. Step6: VerifythatD=P−1APisadiagonalmatrixwhosediientryistheeigenvalueforthe fundamentaleigenvectorformingtheithcolumnofP.AlsonotethatA=PDP−1. Simplified Span Method (Simplifying Span(S)) SupposethatSisafinitesubsetofRn containingkvectors,withk≥2. Tofindasimplifiedformforspan(S),performthefollowingsteps: Step1: Formak×nmatrixAbyusingthevectorsinSastherowsofA.(Thus,span(S)is therowspaceofA.) Step2: LetCbethereducedrowechelonformmatrixforA. Step3: Then,asimplifiedformforspan(S)isgivenbythesetofalllinearcombinationsof thenonzerorowsofC. IndependenceTestMethod (Testing for Linear Independenceof S) LetSbeafinitenonemptysetofvectorsinRn. TodeterminewhetherSislinearlyindependent,performthefollowingsteps: Step1: CreatethematrixAwhosecolumnsarethevectorsinS. Step2: FindB,thereducedrowechelonformofA. Step3: IfthereisapivotineverycolumnofB,thenS islinearlyindependent.Otherwise, Sislinearlydependent. Coordinatization Method (Coordinatizing v with Respectto an OrderedBasis B) Let V be a nontrivial subspaceof Rn, letB = (v ...,v )be an ordered basis for V, 1, k andletv∈Rn.Tocalculate[v] ,ifitexists,performthefollowing: B Step1: Formanaugmentedmatrix[A|v]byusingthevectorsinBasthecolumns ofA, inorder,andusingvasacolumnontheright. Step2: Rowreduce[A|v]toobtainthereducedrowechelonform[C|w]. Step 3: If there is a row of [C|w] that contains all zeroes on the left and has a nonzero entry on the right, then v∈/span(B) = V, and coordinatization is not possible. Stop. Step4: Otherwise,v∈span(B)=V.Eliminateallrowsconsistingentirelyofzeroesin[C|w] toobtain[Ik|y].Then,[v]B=y,thelastcolumnof[Ik|y]. Transition Matrix Method (Calculating a Transition Matrix from B to C) To find the transition matrix P from B to C where B and C are ordered bases for a nontrivialk-dimensionalsubspaceofRn,userowreductionon ⎡ (cid:5) ⎤ 1st 2nd kth (cid:5)(cid:5) 1st 2nd kth ⎢ vector vector ··· vector (cid:5) vector vector ··· vector ⎥ ⎣ (cid:5) ⎦ in in in (cid:5) in in in (cid:5) C C C B B B toproduce (cid:9) (cid:10) I P k . rowsof zeroes Elementary Linear Algebra Elementary Linear Algebra Fifth Edition Stephen Andrilli Department of Mathematics and Computer Science La Salle University Philadelphia, PA David Hecker Department of Mathematics Saint Joseph’s University Philadelphia, PA AMSTERDAM (cid:129) BOSTON (cid:129) HEIDELBERG (cid:129) LONDON NEW YORK (cid:129) OXFORD (cid:129) PARIS (cid:129) SAN DIEGO SAN FRANCISCO (cid:129) SINGAPORE (cid:129) SYDNEY (cid:129) TOKYO Academic Press is an imprint of Elsevier AcademicPressisanimprintofElsevier 125LondonWall,London,EC2Y5AS,UK 525BStreet,Suite1800,SanDiego,CA92101-4495,USA 50HampshireStreet,5thFloor,Cambridge,MA02139,USA TheBoulevard,LangfordLane,Kidlington,OxfordOX51GB,UK Copyright©2016,2010,1999ElsevierInc.Allrightsreserved. Nopartofthispublicationmaybereproducedortransmittedinanyformorbyanymeans,electronic ormechanical,includingphotocopying,recording,oranyinformationstorageandretrievalsystem, withoutpermissioninwritingfromthepublisher.Detailsonhowtoseekpermission,further informationaboutthePublisher’spermissionspoliciesandourarrangementswithorganizationssuch astheCopyrightClearanceCenterandtheCopyrightLicensingAgency,canbefoundatourwebsite: www.elsevier.com/permissions. Thisbookandtheindividualcontributionscontainedinitareprotectedundercopyrightbythe Publisher(otherthanasmaybenotedherein). Notices Knowledgeandbestpracticeinthisfieldareconstantlychanging.Asnewresearchandexperience broadenourunderstanding,changesinresearchmethods,professionalpractices,ormedical treatmentmaybecomenecessary. Practitionersandresearchersmustalwaysrelyontheirownexperienceandknowledgeinevaluating andusinganyinformation,methods,compounds,orexperimentsdescribedherein.Inusingsuch informationormethodstheyshouldbemindfuloftheirownsafetyandthesafetyofothers, includingpartiesforwhomtheyhaveaprofessionalresponsibility. Tothefullestextentofthelaw,neitherthePublishernortheauthors,contributors,oreditors, assumeanyliabilityforanyinjuryand/ordamagetopersonsorpropertyasamatterofproducts liability,negligenceorotherwise,orfromanyuseoroperationofanymethods,products, instructions,orideascontainedinthematerialherein. ISBN:978-0-12-800853-9 BritishLibraryCataloguinginPublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary LibraryofCongressCataloging-in-PublicationData AcatalogrecordforthisbookisavailablefromtheLibraryofCongress ForinformationonallAcademicPresspublications visitourwebsiteathttp://store.elsevier.com/ To our wives, Ene and Lyn, for all their help and encouragement Preface for the Instructor This textbook is intended for a sophomore- or junior-level introductory course in linearalgebra.Weassumethestudentshavehadatleastonecourseincalculus. PHILOSOPHY OF THE TEXT Helpful Transition from Computation to Theory: Our main objective in writing this textbook was to present the basic concepts of linear algebra as clearly as possible. The “heart” of this text is the material in Chapters 4 and 5 (vector spaces, linear transformations). Inparticular,wehavetakenspecialcare toguidestudentsthrough these chapters as the emphasis changes from computation to abstract theory. Many theoretical concepts (such as linear combinations of vectors, the row space of a matrix, and eigenvalues and eigenvectors) are first introduced in the early chapters in order to facilitate a smoother transition to later chapters. Please encourage the studentstoreadthetextdeeplyandthoroughly. Applications of Linear Algebra and Numerical Techniques: This text contains a wide variety of applications of linear algebra, as well as all of the standard numerical techniques typically found in most introductory linear algebra texts. Aside from the many applications and techniques already presented in the first seven chapters, Chapter 8 is devoted entirely to additional applications, while Chapter 9 introduces severalothernumericaltechniques.Asummaryoftheseapplicationsandtechniques isgiveninthechartlocatedattheendofthePrefaces. Numerous Examples and Exercises: There are 340 numbered examples in the text, at leastoneforeachmajorconceptorapplication,aswellasforalmosteverytheorem. The text also contains an unusually large number of exercises. There are more than 970 numbered exercises, and many of these have multiple parts, for a total of more than 2600 questions. The exercises within each section are generally ordered by increasingdifficulty,beginningwithbasiccomputationalproblemsandmovingonto more theoretical problems and proofs. Answers are provided at theend ofthe book for approximately half of the computational exercises; these problems are marked withastar((cid:2)).FullsolutionstothesestarredexercisesappearintheStudentSolutions Manual. The last exercises in each section are True/False questions (there are over 500 of these altogether). These are designed to test the students’ understanding of fundamentalconceptsbyemphasizingtheimportanceofcriticalwordsindefinitions or theorems. Finally, there is a set of comprehensive Review Exercises at the end of eachofChapters1through7. AssistanceintheReadingandWritingofMathematical Proofs: Topreparestudentsforan increasing emphasis on abstract concepts, we introduce them to proof-reading and xi xii Preface for the Instructor proof-writing very early in the text, beginning with Section 1.3, which is devoted solelytothistopic.Forlongproofs,wepresentstudentswithanoverviewsotheydo notgetlostinthedetails.Foreverynontrivial theoreminChapters1through6,we have either included a proof, or given detailedhints to enable students to provide a proof ontheirown. Mostof theproofs left asexercisesare marked withthe symbol (cid:2),andtheseproofscanbefoundintheStudentSolutionsManual. Symbol Table: Following the Prefaces, for convenience, there is a comprehensive Symbol Table summarizing all of the major symbols for linear algebra that are employedinthistext. Instructor’s Manual: An Instructor’s Manual is available online for all instructors who adoptthistext.Thismanualcontainstheanswerstoallexercises,bothcomputational andtheoretical.ThismanualalsoincludesthreeversionsofaSampleTestforeachof Chapters1through7,alongwithcorrespondinganswerkeys. Student Solutions Manual: A Student Solutions Manual is available for students to purchase online. This manual contains full solutions for each exercise in the text bearing a (cid:2) (those whose answers appear in Appendix E). The Student Solutions Manual also contains the proofs of most of the theorems that were left to the exercises. These exercises are marked in the text with a (cid:2). Because we have compiled this manual ourselves, it utilizes the same styles of proof-writing and solutiontechniquesthatappearintheactualtext. AdditionalMaterialontheWeb:Thewebsiteforthistextbookis: http://store.elsevier.com/9780128008539 This site contains information about the text, as well as several sections of subject contentthatareavailableforinstructorsandstudentswhoadoptthetext.Theseweb sectionsrangefromelementarytoadvancedtopics–fromtheCrossProducttoJordan CanonicalForm.Thesecanbecoveredbyinstructorsintheclassroom,orusedbythe studentstoexploreontheirown. MAJOR CHANGES FOR THE FIFTH EDITION Welistsomeofthemostimportant specificchangessectionbysectionhere,butfor brevity’s sake, we will not detail every specific change. However,some of the more generalsystemicrevisionsinclude: (cid:2) Some reordering of theorems and examples was done in various chapters (particularly in Chapters 2 through5) tostreamline the presentation of results (see, e.g., Theorem 2.2 and Corollary 2.3, Theorem 4.9, Theorem 5.16, Corol- lary6.16,Theorem6.21,Corollary6.23,andCorollary7.21). Preface for the Instructor xiii (cid:2) Variousimportant,frequentlyused,statementsinthetexthavebeenconverted into formal theorems for easier reference in proofs and solutions to exercises (e.g.,inSections6.2,7.5,and8.8). (cid:2) The Highlights appearing at the conclusion of each section have been revised substantially to be more concrete in nature so that they are more useful to studentslookingforaquickreviewofthesection. (cid:2) Many formatting changes were made in the textbook for greater readability to displaysomeintermediatestepsandfinalresultsofsolutionsmoreprominently. Othersectionsthatreceivedsubstantialchangesfromthefourtheditionare: (cid:2) Section1.1 (andbeyond): Thenotionofa“GeneralizedEtchASketch(cid:3)”1 has been introduced, beginning in Section 1.1, to help students betterunderstand thevectorconceptsoflinearcombinations,span,andlinearindependence. (cid:2) Section 3.3: The text now includes a proof for (the current) Theorem 3.10, eliminatingtheneedforalongsequenceofintertwinedexercisesthatformerly concludedthissection.Also,thematerialontheclassicaladjointofamatrixhas beenmovedtotheexercises. (cid:2) Section 4.5: The proof of (the current) Lemma 4.10 has been shortened. In ordertosharpenthefocusonmoreessentialconcepts,thematerialonmaximal linearlyindependentsubsetsandminimalspanningsubsetshasbeeneliminated. This necessitateda change in the proof that a subspace of a finite-dimensional vectorspaceisfinitedimensional.Section4.6wasalsoadjustedaccordingly. (cid:2) Section 4.6: The Inspection Method (for reducing a finite spanning set to a basis)hasbeenmovedtotheexercises. (cid:2) Section 5.4: The result now stated as Corollary 5.13 was moved here from Section5.5. (cid:2) Section8.1:ThissectiononGraphTheorywasthoroughlyupdated,withmore generalgraphsanddigraphstreated,ratherthanjustsimplegraphsanddigraphs. Also,newmaterialhasbeenaddedontheconnectivityofgraphs. (cid:2) Section8.3:Theconceptof“interpolation”hasbeenintroduced. (cid:2) Appendix B: This appendix on basic properties of functions was thoroughly rewrittentomaketheoremsandexamplesmorevisible. (cid:2) AppendixC:Statementsofthecommutative,associative,anddistributivelaws foradditionandmultiplicationofcomplexnumbershavebeenadded. (cid:2) Appendix D: The section “Elementary Matrices” (formerly Section 8.6 in the fourth edition) has been moved to Appendix D. Consequently,the answers to starredexercisesnowappearinAppendixE. 1EtchASketch(cid:3)isaregisteredtrademarkoftheOhioArtCompany.

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Elementary Linear. Algebra. Fifth Edition. Stephen Andrilli. Department of Mathematics and Computer Science. La Salle University. Philadelphia, PA. David Hecker. Department of Mathematics. Saint Joseph's University. Philadelphia, PA. AMSTERDAM • BOSTON • HEIDELBERG • LONDON. NEW YORK
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