Elementary Linear Algebra (2016)

794 Pages·2016·10.35 MB·English

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
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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.