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Fibonacci vectors

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Fibonacci vectors
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Salter, Ena
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University of South Florida
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Linear algebra
Lucas numbers
Product identites
Asymptotics
Golden ratio
Dissertations, Academic -- Mathematics -- Masters -- USF   ( lcsh )
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government publication (state, provincial, terriorial, dependent)   ( marcgt )
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Abstract:
ABSTRACT: By the n-th Fibonacci (respectively Lucas) vector of length m, we mean the vector whose components are the n-th through (n+m-1)-st Fibonacci (respectively Lucas) numbers. For arbitrary m, we express the dot product of any two Fibonacci vectors, any two Lucas vectors, and any Fibonacci vector and any Lucas vector in terms of the Fibonacci and Lucas numbers. We use these formulas to deduce a number of identities involving the Fibonacci and Lucas numbers.
Thesis:
Thesis (M.A.)--University of South Florida, 2005.
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Includes bibliographical references.
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by Ena Salter.
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Document formatted into pages; contains 46 pages.
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FibonacciVectorsbyEnaSalterAthesissubmittedinpartialfulllmentoftherequirementsforthedegreeofMasterofArtsDepartmentofMathematicsCollegeofArtsandSciencesUniversityofSouthFloridaMajorProfessor:BrianCurtin,PhD.ThomasBieske,PhD.DavidStone,PhD.DateofApproval:July20,2005Keywords:linearalgebra,Lucasnumbers,productidentites,asymptotics,goldenratiocCopyright2005,Salter

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DedicationForthethreemajorrolemodelsinmylife.Mymomwhosecourageandstrengthmakemesoproud.Mydadwhoseunconditionalloveismyguidinglight.Andforthebestteacherintheworld,Dr.Stone.

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AcknowledgementsTherearesomanypeopletowhomIoweagreatdealofappreciation,forwithoutthemIwouldnotIhavemadeitthusfar.Firsto,IwouldliketothankthemanymathprofessorsIhadatGeorgiaSouthernfortheenormousamountoftimetheydevotedtoteachingmethatmathwasmorethanjustnumbersandletters.Secondly,IwouldliketothankDr.BrianCurtinforspendingcountlesshourshelpingmenotonlyunderstandmathematicsbutalsolearnhowtoprograminTEX,hispatiencewillneverbeforgotten.IwouldalsoliketothankDr.ThomasBieskeforagreeingtobeonmydefensecommittee.Ofcourse,IwouldalsoliketothankDr.DavidStoneforseeinginmemydesiretolearnandinspiringmetobecomeamathteacher.Also,thankstomymotherforteachingmethemostvaluablelessoninlife...tonevergiveupnomatterhowroughtheroadmaybe.Andabigthankyoutomydaddyforhisendlessamountofsupportbothnainciallyandemotionally.WithouttheloveofmyparentsandmybrotherJamesIwouldhaveneverbeenabletoaccomplishsomuchinmylife.Finally,thankstoallofmyfriendsMindy,Meagan,Erin,Charlotte,Daina,Joey,John,andMichaelforlisteningtomerantandputtingupwithmewhenImusthavebeenimpossibletobearound.Youguysaretrulyablessinginmylifeandyoursupportshowsthroughoutmythesis.IloveallofyouandIamsothankfultocallyouguysmyfriends.Throughtopology,thebadboyfriends,thenrandomallergicreactions,theminorandmajorillnessesIcouldalwayscountonyouMegas.Wedidit!

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TableofContentsAbstractiii1Introduction12FibonacciandLucasnumbers32.1FundamentalFibonacciandLucasfacts................. 3 2.2Alinearalgebraicperspective...................... 4 2.3TheBinetformulas............................ 5 3FibonacciandLucasvectors103.1Linearalgebraicsetup.......................... 10 3.2FibonacciandLucasvectors....................... 13 3.3Dotproducts............................... 14 3.4FibonacciandLucasidentities...................... 17 4Anglesinevendimension224.1Dotproductswith~aand~b........................ 22 4.2Cosinesofangleswith~aand~b...................... 23 4.3Anglesbetweenvectors.......................... 25 4.4Angleswiththeaxes........................... 27 4.5Dimensiontwo.............................. 27 5Commentsonanglesinodddimension305.1Theanglebetween~aand~b........................ 30 5.2Three-dimensionalFibonacciVectors.................. 31 i

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6Asymptoticsofangles336.1Angleswith~a............................... 33 6.2Dominanteigenvalues........................... 34 6.3Furtherdirections............................. 36 References39AbouttheAuthorEndPage ii

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FibonacciVectorsEnaSalterABSTRACTBythen-thFibonaccirespectivelyLucasvectoroflengthm,wemeanthevectorwhosecomponentsarethen-ththroughn+m)]TJ/F19 11.955 Tf 10.835 0 Td[(1-stFibonaccirespectivelyLucasnumbers.Forarbitrarym,weexpressthedotproductofanytwoFibonaccivectors,anytwoLucasvectors,andanyFibonaccivectorandanyLucasvectorintermsoftheFibonacciandLucasnumbers.WeusetheseformulastodeduceanumberofidentitiesinvolvingtheFibonacciandLucasnumbers. iii

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1IntroductionSeldom,inthestudyofmathematics,doesonecomeacrossatopicsofascinatingthatitcaptivatesthemindsofmathematiciansandnon-mathematiciansalike.TheFibonaccisequence,thoughover700yearsold,stillcontainsmanysecretsyettobediscovered.ThefamoussequencehasintriguedsomanypeoplethatTheFibonacciQuarterly,ajournaldevotedsolelytothestudyofanythingFibonacciwascreatedin1963,shortlyaftertheformationoftheFibonacciSocietyin1962.Veryfamousduringhislife,FibonacciisconsideredthegreatestEuropeanmathe-maticianofthemiddleages.Fibonacci,shortforFiliusBonacci{sonofBonacci{wasbornaround1170,toaPisanMerchantwhofreelytraveledtheexpanseoftheByzan-tineEmpire.Duetotheextensivetraveling,LeonardoofPisawasfrequentlyexposedtoIslamicscholarsandthemathematicsoftheIslamicworld.AfterhisreturntoPisa,Fibonaccispentthenext25yearswritingbooksthatincludedmuchofwhathehadlearnedinhistravels.Thoughmanyworkswerelost,threemainworkswerepreserved.Theyare:Liberabaci202,1228,thePracticageometriae220,andtheLiberquadratorum225.FibonacciiscreditedwithbeingoneoftherstpeopletointroducetheHindu-ArabicnumbersystemintoEurope{thesystemwenowusetoday{basedoftendigitswithitsdecimalpointandasymbolforzero:1,2,3,4,5,6,7,8,9and0.ItisinLiberabacithatFibonacciposeshisfamousrabbitquestion, Acertainmanputapairofrabbitsinaplacesurroundedonallsidesbyawall.Howmanypairsofrabbitscanbeproducedfromthatpairinayearifitissupposedthateverymontheachpairbegetsanewpairwhichfrom 1

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thesecondmonthonbecomesproductive?TheanswertothisquestioninvolvesthefamousFibonaccisequence:1,1,2,3,5,8,13,...wherethen-thFibonaccinumber,denotedFn,isdenedforallintegersnbytherecurrencerelationFn+2=Fn+1+Fn,withstartingvaluesF1=F2=1.Whatseemslikesuchasimpleconceptappearsinalmosteveryscienticeldofstudyfrombotanytoarchitecture,andbiologytopainting.WearebutbeingsswimmingintheseaofFibonacciwhereallwemustperformisasinglestroketomeetaFibonaccinumber.ItisthissequenceofnumbersthatinspiresalloftheideasinthispaperandtonosuprisewendthatwheneveroneworkswithFibonaccinumbersweobtainbeautifulresults.Wewilllookatvectorsoftheform~fn=hFn;Fn+1;Fn+2;:::;Fn+m)]TJ/F25 7.97 Tf 6.587 0 Td[(1i,andcallthemFibonaccivectors.WewillstudyindepththeFibonaccivectorsinarbitrarydimensions.Simultaneously,weshallconsideranalagousresultsforLucasvectors.EdouardLucaswasborninFrancein1842.FollowingserviceasanartilleryocerduringtheFranco-PrussianWar870-1871,LucasbecameprofessorofmathematicsattheLyceeSaintLouisinParis.HelaterbecameprofessorofmathematicsattheLyceeCharlemagne,alsoinParis.Lucasisbestknownforhisstudiesinnumbertheory.WewillstudytheLucassequence:1,3,4,7,11,18,29,...wherethen-thLucasnumber,denotedLn,isdenedforallintegersnbytherecurrencerelationLn+2=Ln+1+Ln,withstartingvaluesL1=1,L2=3.WelookatanalogousLucasvectorsoftheform~`n=hLn;Ln+1;Ln+2;:::;Ln+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1i.Wewillbeginourstudybyrecallingsomewell-knownfactsconcerningtheFi-bonacciandLucasvectorsintwodimensions.Wethengeneralizethesefactstoarbitrarydimension.FromveryelementarylinearalgebraicconsiderationsweshallderiveanumberofnontrivialrelationsinvolvingtheFibonacciandLucasnumbers.Itturnsoutthatvectorsinaxeddimensionlieinasingleplane.Weconsideranglesbetweenthevectorsofinterest. 2

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2FibonacciandLucasnumbers 2.1FundamentalFibonacciandLucasfactsTheFibonaccinumbersFnaredenedforallintegersnbythesecondorderrecurrencerelationFn+2=Fn+1+Fn .1.1 andinitialconditionsF1=F2=1: .1.2 TheLucasnumbersLnaredenedforallintegersnbythesamesecondorderrecur-rencerelationastheFibonaccinumbersLn+2=Ln+1+Ln .1.3 butinitialconditionsL1=1;L2=3: .1.4 WerecallsomerelationsinvolvingtheFibonacciandLucasnumbers.Thesefactsarewell-knownandcanbefoundinmostbasicreferences,e.g.[3,4]. Theorem2.1.1[3] Forallintegersn1,F2n=F2n+1)]TJ/F27 11.955 Tf 11.956 0 Td[(F2n)]TJ/F25 7.97 Tf 6.586 0 Td[(1; .1.5 3

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F2n)]TJ/F25 7.97 Tf 6.587 0 Td[(1F2n+1=F22n+1; .1.6 Fn=FkFn)]TJ/F28 7.97 Tf 6.587 0 Td[(k+1+Fk)]TJ/F25 7.97 Tf 6.586 0 Td[(1Fn)]TJ/F28 7.97 Tf 6.586 0 Td[(k; .1.7 F2n+1=F2n+F2n+1; .1.8 Fn=Ln)]TJ/F25 7.97 Tf 6.586 0 Td[(1+Ln+1; .1.9 F)]TJ/F28 7.97 Tf 6.587 0 Td[(n=)]TJ/F19 11.955 Tf 9.299 0 Td[(1n+1Fn; .1.10 L)]TJ/F28 7.97 Tf 6.587 0 Td[(n=)]TJ/F19 11.955 Tf 9.299 0 Td[(1nLn: .1.11 2.2AlinearalgebraicperspectiveWerecallawell-knownlinearalgebraicapproachtotheFibonaccinumbers.Thiscanbefoundin[4]andmanyelementarylinearalgebrabooks,e.g.[7,9].ThisapproachappearstobeduetoBinet.SetT=24011135: .2.12 Theorem2.2.1[4] LetTbeasinEqn..2.12.Forallintegersn,let~fn=[Fn;Fn+1]t.Then~fn+1=T~fn:Inparticular,forallintegersk~fn+1=Tn)]TJ/F28 7.97 Tf 6.586 0 Td[(k+1~fk: .2.13 Theorem2.2.1canbeusedtoproveanumberofFibonacciidentities.ObservethattherowsandcolumnsofTaresimply~f0and~f1,sothatTn+1hasrowsandcolumns~fn)]TJ/F25 7.97 Tf 6.586 0 Td[(1and~fn.Thus,thesimpleobservationthatTn=Tn)]TJ/F28 7.97 Tf 6.586 0 Td[(kTkgivestheidentity.1.7.See[4].Recently,Askey[1,2]andHuang[5]havegivenmatrixtheoreticproofsinthisspiritofotherFibonacciidentitiesasaninterestingapplicationofmatrixmultiplication. 4

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Ourworkappliesthissortofargumentinarbitrarydimension.Ratherthanusematrixmultiplication,weconsiderthedotproductofvectorswithcomponentscon-sistingofconsecutiveFibonaccinumbers.Thisisanaturalgeneralizationsincetheentriesinamatrixproductcanbeviewedasadotproductofarowwithacolumn.WeshalltreattheLucasvectorsinthesamemanner. 2.3TheBinetformulasWerecallclosed-formformulasfortheFibonacciandLucasnumbersknownastheBinetFormulas.TheseformulasareoftenfoundasanapplicationofTheorem2.2.1inelementarylinearalgebratextbooks.Werecalltheseformulasandthecommonlinearalgebraicderivationnow. Lemma2.3.1 ThematrixTofEqn..2.12hascharacteristicpolynomialx2)]TJ/F27 11.955 Tf 10.066 0 Td[(x)]TJ/F19 11.955 Tf 10.065 0 Td[(1.Thusithaseigenvalues=1+p 5 2and=1)]TJ 6.587 6.598 Td[(p 5 2.Theassociatedeigenvectorsarerespectively24135;24135: Lemma2.3.2 Forallintegersn,~fn=1 )]TJ/F27 11.955 Tf 11.955 0 Td[(0@n24135)]TJ/F27 11.955 Tf 11.955 0 Td[(n241351A;~`n=n24135+n24135:Proof.Observethat~f0=240135=1 )]TJ/F27 11.955 Tf 11.955 0 Td[(0@024135)]TJ/F27 11.955 Tf 11.955 0 Td[(0241351A;~`0=242135=024135+024135: 5

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Now,~fn=Tn~f0=1 )]TJ/F27 11.955 Tf 11.955 0 Td[(0@Tn24135)]TJ/F27 11.955 Tf 11.956 0 Td[(Tn241351AbyEqn..2.13=1 )]TJ/F27 11.955 Tf 11.955 0 Td[(0@n24135)]TJ/F27 11.955 Tf 11.955 0 Td[(n241351AbyLem.2.3.1: Theorem2.3.3 Forallintegersn,Fn=n)]TJ/F27 11.955 Tf 11.955 0 Td[(n )]TJ/F27 11.955 Tf 11.955 0 Td[(; .3.14 Ln=n+n: .3.15 Proof.EquatetherstcomponentsoneachsideoftheseequationsofLemma2.3.2. Equations.3.14and.3.15aregenerallyknownastheBinetformulasforFnandLnalthoughpreviouslyknowntoEulerandDanielBernoulli[4].WeshallmakeextensiveuseoftheBinetformulas,sowepresentsomeformulasinvolvingandandsomealternateversionsoftheBinetformulas. Lemma2.3.4 =)]TJ/F19 11.955 Tf 9.298 0 Td[(1; .3.16 +=1; .3.17 )]TJ/F27 11.955 Tf 11.955 0 Td[(=p 5; .3.18 2+1=p 5; .3.19 2+1=)]TJ 9.298 10.473 Td[(p 5; .3.20 =2)]TJ/F27 11.955 Tf 11.955 0 Td[(2; .3.21 =2)]TJ/F27 11.955 Tf 11.955 0 Td[(2: .3.22 6

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Proof.Immediatesinceandaretherootsofx2)]TJ/F27 11.955 Tf 11.955 0 Td[(x)]TJ/F19 11.955 Tf 11.955 0 Td[(1. Lemma2.3.5 [4]Forallintegersn,n=Fn+Fn)]TJ/F25 7.97 Tf 6.586 0 Td[(1; .3.23 n=Fn+Fn)]TJ/F25 7.97 Tf 6.587 0 Td[(1: .3.24 Lemma2.3.6 Forallintegersn1andn2,n1n2+n2n1=)]TJ/F19 11.955 Tf 9.299 0 Td[(1n1Ln2)]TJ/F28 7.97 Tf 6.587 0 Td[(n1; .3.25 n1n2)]TJ/F27 11.955 Tf 11.955 0 Td[(n2n1=)]TJ/F19 11.955 Tf 9.299 0 Td[(1n1+1)]TJ/F27 11.955 Tf 11.955 0 Td[(Fn2)]TJ/F28 7.97 Tf 6.586 0 Td[(n1: .3.26 Proof.Observethatn1n2=1 2n1n2+n1n2=1 2)]TJ/F19 11.955 Tf 5.479 -9.683 Td[()]TJ/F19 11.955 Tf 9.298 0 Td[(1n1n2)]TJ/F28 7.97 Tf 6.587 0 Td[(n1+)]TJ/F19 11.955 Tf 9.299 0 Td[(1n2n1)]TJ/F28 7.97 Tf 6.586 0 Td[(n2byEqn..3.16;n2n1=1 2n2n1+n2n1=1 2)]TJ/F19 11.955 Tf 5.479 -9.684 Td[()]TJ/F19 11.955 Tf 9.298 0 Td[(1n2n1)]TJ/F28 7.97 Tf 6.587 0 Td[(n2+)]TJ/F19 11.955 Tf 9.299 0 Td[(1n1n2)]TJ/F28 7.97 Tf 6.586 0 Td[(n1byEqn..3.16:Thusn1n2+n2n1=1 2)]TJ/F19 11.955 Tf 5.48 -9.684 Td[()]TJ/F19 11.955 Tf 9.299 0 Td[(1n1n2)]TJ/F28 7.97 Tf 6.586 0 Td[(n1+n2)]TJ/F28 7.97 Tf 6.586 0 Td[(n1+)]TJ/F19 11.955 Tf 9.299 0 Td[(1n2n1)]TJ/F28 7.97 Tf 6.586 0 Td[(n2+n1)]TJ/F28 7.97 Tf 6.587 0 Td[(n2=)]TJ/F19 11.955 Tf 9.298 0 Td[(1n1Ln2)]TJ/F28 7.97 Tf 6.586 0 Td[(n1+)]TJ/F19 11.955 Tf 9.298 0 Td[(1n2Ln1)]TJ/F28 7.97 Tf 6.586 0 Td[(n2 2byEqn..3.15=)]TJ/F19 11.955 Tf 9.299 0 Td[(1n1Ln2)]TJ/F28 7.97 Tf 6.587 0 Td[(n1byEqn..1.11=)]TJ/F19 11.955 Tf 9.299 0 Td[(1n2Ln1)]TJ/F28 7.97 Tf 6.587 0 Td[(n2byEqn..1.11:Similarly,n1n2)]TJ/F27 11.955 Tf 11.956 0 Td[(n2n1=1 2)]TJ/F19 11.955 Tf 5.479 -9.683 Td[()]TJ/F19 11.955 Tf 9.298 0 Td[(1n1)]TJ/F27 11.955 Tf 9.299 0 Td[(n2)]TJ/F28 7.97 Tf 6.586 0 Td[(n1+n2)]TJ/F28 7.97 Tf 6.586 0 Td[(n1+)]TJ/F19 11.955 Tf 9.299 0 Td[(1n2n1)]TJ/F28 7.97 Tf 6.586 0 Td[(n2)]TJ/F27 11.955 Tf 11.955 0 Td[(n1)]TJ/F28 7.97 Tf 6.587 0 Td[(n2 7

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=)]TJ/F27 11.955 Tf 11.955 0 Td[( 2)]TJ/F19 11.955 Tf 5.48 -9.684 Td[()]TJ/F19 11.955 Tf 9.299 0 Td[(1n1+1Fn2)]TJ/F28 7.97 Tf 6.586 0 Td[(n1+)]TJ/F19 11.955 Tf 9.299 0 Td[(1n2Fn1)]TJ/F28 7.97 Tf 6.587 0 Td[(n2byEqn..3.14=)]TJ/F19 11.955 Tf 9.298 0 Td[(1n1+1)]TJ/F27 11.955 Tf 11.955 0 Td[(Fn2)]TJ/F28 7.97 Tf 6.586 0 Td[(n1byEqn..1.10=)]TJ/F19 11.955 Tf 9.298 0 Td[(1n2)]TJ/F27 11.955 Tf 11.955 0 Td[(Fn1)]TJ/F28 7.97 Tf 6.587 0 Td[(n2byEqn..1.10: Corollary2.3.7 Forallintegersn, n+ n=)]TJ/F19 11.955 Tf 9.298 0 Td[(1nL2n; n)]TJ/F32 11.955 Tf 11.955 16.857 Td[( n=)]TJ/F19 11.955 Tf 9.298 0 Td[(1n)]TJ/F27 11.955 Tf 11.956 0 Td[(F2n:Proof.Taken1=nandn2=)]TJ/F27 11.955 Tf 9.299 0 Td[(nin.3.25and2.3.26. Lemma2.3.8 Forallintegersn,Fn=1 n)]TJ/F25 7.97 Tf 6.587 0 Td[(12n)]TJ/F19 11.955 Tf 11.955 0 Td[()]TJ/F19 11.955 Tf 9.299 0 Td[(1n 2)]TJ/F19 11.955 Tf 11.956 0 Td[(1 .3.27 =1 n)]TJ/F25 7.97 Tf 6.586 0 Td[(12n)]TJ/F19 11.955 Tf 11.955 0 Td[()]TJ/F19 11.955 Tf 9.299 0 Td[(1n 2)]TJ/F19 11.955 Tf 11.956 0 Td[(1; .3.28 Ln=2n+)]TJ/F19 11.955 Tf 9.298 0 Td[(1n n .3.29 =2n+)]TJ/F19 11.955 Tf 9.299 0 Td[(1n n: .3.30 Proof.ComputeFn=n)]TJ/F27 11.955 Tf 11.955 0 Td[(n )]TJ/F27 11.955 Tf 11.955 0 Td[(byEqn..3.14=2n)]TJ/F27 11.955 Tf 11.955 0 Td[(nn 2)]TJ/F27 11.955 Tf 11.955 0 Td[( n=1 n)]TJ/F25 7.97 Tf 6.586 0 Td[(12n)]TJ/F19 11.955 Tf 11.956 0 Td[()]TJ/F19 11.955 Tf 9.298 0 Td[(1n 2)]TJ/F19 11.955 Tf 11.955 0 Td[(1byEqn..3.16;Ln=n+n 8

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=2n+nn n=2n+)]TJ/F19 11.955 Tf 9.299 0 Td[(1n nEquations2.3.29and.3.30arederivedsimilarly. 9

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3FibonacciandLucasvectorsInthischapterwestudyFibonacciandLucasvectorsinarbitrarydimensionfromalinearalgebraicperspective. 3.1LinearalgebraicsetupThroughoutthissectionmshallbeaxedpositiveinteger.DeneanmmmatrixTbyT=26666666666640100000100..................0001000001000113777777777775: .1.1 Lemma3.1.1 ThematrixTofEqn..1.1hascharacteristicpolynomialxm)]TJ/F25 7.97 Tf 6.586 0 Td[(1x2)]TJ/F27 11.955 Tf -424.076 -20.921 Td[(x)]TJ/F19 11.955 Tf 11.703 0 Td[(1.Thusthenon-zeroeigenvaluesofTare=+p 5=2and=)]TJ 11.702 9.889 Td[(p 5=2,eachwithgeometricandalgebraicmultiplicity1.Theeigenspacesassociatedwith 10

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andarespannedrespectivelyby~a=266666666412...m)]TJ/F25 7.97 Tf 6.587 0 Td[(13777777775and~b=266666666412...m)]TJ/F25 7.97 Tf 6.587 0 Td[(13777777775:Proof.Elementarylinearalgebra. ObservethatthematrixTandvectors~aand~bdependuponthedimensionm.Sincemwillalwaysbeclearfromcontext,weshallsuppressthisdependenceinnotation.Weshallfrequentlyusetheformulaforthesumofanitegeometricseries. Lemma3.1.2 Forallrealnumbersc6=1andallpositiveintegersm,m)]TJ/F25 7.97 Tf 6.586 0 Td[(1Xj=0cj=cm)]TJ/F19 11.955 Tf 11.955 0 Td[(1 c)]TJ/F19 11.955 Tf 11.955 0 Td[(1: .1.2 Lemma3.1.3 ~a~a=Fm)]TJ/F27 11.955 Tf 11.955 0 Td[(m)]TJ/F25 7.97 Tf 6.587 0 Td[(1ifmiseven,Lmm)]TJ/F25 7.97 Tf 6.587 0 Td[(1ifmisodd, .1.3 ~b~b=)]TJ/F27 11.955 Tf 9.298 0 Td[(Fm)]TJ/F27 11.955 Tf 11.955 0 Td[(m)]TJ/F25 7.97 Tf 6.587 0 Td[(1ifmiseven,Lmm)]TJ/F25 7.97 Tf 6.586 0 Td[(1ifmisodd, .1.4 ~a~b=0ifmiseven,1ifmisodd. .1.5 Proof.Bydenitionofdotproduct~a~a=m)]TJ/F25 7.97 Tf 6.587 0 Td[(1Xj=02j=m)]TJ/F25 7.97 Tf 6.587 0 Td[(1Xj=0)]TJ/F27 11.955 Tf 9.299 0 Td[( jbyEqn..3.16 11

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=)]TJ/F27 11.955 Tf 9.298 0 Td[(=m)]TJ/F19 11.955 Tf 11.955 0 Td[(1 )]TJ/F27 11.955 Tf 9.298 0 Td[(=)]TJ/F19 11.955 Tf 11.956 0 Td[(1byEqn..1.2=1 m)]TJ/F25 7.97 Tf 6.587 0 Td[(1)]TJ/F19 11.955 Tf 9.299 0 Td[(1mm)]TJ/F27 11.955 Tf 11.955 0 Td[(m )]TJ/F27 11.955 Tf 9.298 0 Td[()]TJ/F27 11.955 Tf 11.955 0 Td[(=m)]TJ/F25 7.97 Tf 6.587 0 Td[(1m)]TJ/F19 11.955 Tf 11.955 0 Td[()]TJ/F19 11.955 Tf 9.299 0 Td[(1mmbyLem.2.3.4:Supposemiseven.Then~a~a=m)]TJ/F25 7.97 Tf 6.587 0 Td[(1)]TJ/F27 11.955 Tf 11.955 0 Td[(m)]TJ/F27 11.955 Tf 11.955 0 Td[(m )]TJ/F27 11.955 Tf 11.955 0 Td[(byEqn..3.16=m)]TJ/F25 7.97 Tf 6.587 0 Td[(1)]TJ/F27 11.955 Tf 11.955 0 Td[(FmbyEqn..3.14:Nowsupposemisodd.Then~a~a=m)]TJ/F25 7.97 Tf 6.587 0 Td[(1m+mbyEqn..3.16=m)]TJ/F25 7.97 Tf 6.587 0 Td[(1LmbyEqn..3.15:Thecomputationof~b~bissimilar,soweomitit.Bydenitionofdotproduct~a~b=m)]TJ/F25 7.97 Tf 6.587 0 Td[(1Xj=0jj=m)]TJ/F25 7.97 Tf 6.587 0 Td[(1Xj=0)]TJ/F19 11.955 Tf 9.299 0 Td[(1jbyEqn..3.16=0ifmiseven,1ifmisodd. Recallthatforanyvector~v,thesquareofthelengthof~visk~vk2=~v~v: 12

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Corollary3.1.4 k~ak=p Fm)]TJ/F27 11.955 Tf 11.955 0 Td[(m)]TJ/F25 7.97 Tf 6.587 0 Td[(1ifmiseven,p Lmm)]TJ/F25 7.97 Tf 6.587 0 Td[(1ifmisodd, .1.6 ~b=p )]TJ/F27 11.955 Tf 9.299 0 Td[(Fm)]TJ/F27 11.955 Tf 11.955 0 Td[(m)]TJ/F25 7.97 Tf 6.587 0 Td[(1ifmiseven,p Lmm)]TJ/F25 7.97 Tf 6.587 0 Td[(1ifmisodd, .1.7 k~ak~b=Fm)]TJ/F27 11.955 Tf 11.955 0 Td[(ifmiseven,Lmifmisodd, .1.8 3.2FibonacciandLucasvectors Denition3.2.1 Forallpositiveintegersmandforallintegersn,dene~fmn=26666664FnFn+1...Fn+m)]TJ/F25 7.97 Tf 6.586 0 Td[(137777775;~`mn=26666664LnLn+1...Ln+m)]TJ/F25 7.97 Tf 6.586 0 Td[(137777775:Wereferto~fmnand~`mnasthen-thFibonacciandLucasvectorsoflengthm,respec-tively.Thevectors~fnand~`nofSection2.2arejust~f2nand~`2ninthepresentnotation.Weshallcarrythedimensionminthenotationasweshallhaveoccasionbelowtousemorethanonevalueofminthesameequation.AnumberofotherrelationsamongtheFibonacciandLucasnumbersgeneralizetothecorrespondingvectors.Observethatthe~fmnand~`mnsatisfythevectorrecurrencerelation~xn+2=~xn+1+~xn:TheanalogofTheorem2.2.1isthefollowing. Lemma3.2.2 LetTbeasinEqn..1.1.Thenforallintegersn,~fmn+1=T~fmn; 13

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~`mn+1=T~`mn:Thusforallintegerskn+1,~fmn+1=Tn)]TJ/F28 7.97 Tf 6.587 0 Td[(k+1~fmk;~`mn+1=Tn)]TJ/F28 7.97 Tf 6.587 0 Td[(k+1~`mk:VectorversionsoftheBinetformulasholdhere. Theorem3.2.3 Forallintegersn,~fmn=1 )]TJ/F27 11.955 Tf 11.955 0 Td[(n~a)]TJ/F27 11.955 Tf 11.955 0 Td[(n~b; .2.9 ~`mn=n~a+n~b: .2.10 Inparticular,~fmnand~`mnlieintheplanespannedby~aand~b.Proof.Thej-thentryof~fmnisFn+j)]TJ/F25 7.97 Tf 6.587 0 Td[(1=n+j)]TJ/F25 7.97 Tf 6.587 0 Td[(1)]TJ/F27 11.955 Tf 10.461 0 Td[(n+j)]TJ/F25 7.97 Tf 6.587 0 Td[(1=)]TJ/F27 11.955 Tf 10.461 0 Td[(.Thej-thentryofn~a)]TJ/F27 11.955 Tf 11.955 0 Td[(n~b=)]TJ/F27 11.955 Tf 11.765 0 Td[(isnj)]TJ/F25 7.97 Tf 6.586 0 Td[(1)]TJ/F27 11.955 Tf 11.765 0 Td[(nj)]TJ/F25 7.97 Tf 6.586 0 Td[(1=)]TJ/F27 11.955 Tf 11.765 0 Td[(.Therstequationfollowssincebothsideshavethesameentries.Asimilarargumentgivesthesecondequation. TheanalogofLemma2.3.5isthefollowing. Lemma3.2.4 Forallintegersn,n~a=~fmn+~fmn)]TJ/F25 7.97 Tf 6.586 0 Td[(1;n~b=~fmn+~fmn)]TJ/F25 7.97 Tf 6.586 0 Td[(1:Proof.ComputecorrespondingentriesoneachsideoftheequationsandnotethattheyareequalbyLemma2.3.5. 3.3DotproductsWecomputethedotproductsofFibonacciandLucasvectors. 14

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Theorem3.3.1 Forallpositiveintegersmandforallintegersn1andn2,~fmn1~fmn2=FmFn1+n2+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1ifmiseven,1 5LmLn1+n2+m)]TJ/F25 7.97 Tf 6.587 0 Td[(1)]TJ/F19 11.955 Tf 11.955 0 Td[()]TJ/F19 11.955 Tf 9.299 0 Td[(1n1Ln2)]TJ/F28 7.97 Tf 6.587 0 Td[(n1ifmisodd. .3.11 Proof.~fmn1~fmn2=1 )]TJ/F27 11.955 Tf 11.955 0 Td[(n1~a)]TJ/F27 11.955 Tf 11.955 0 Td[(n1~b1 )]TJ/F27 11.955 Tf 11.955 0 Td[(n2~a)]TJ/F27 11.955 Tf 11.955 0 Td[(n2~bbyEqn..2.9=1 5n1+n2~a~a+n1+n2~b~b)]TJ/F19 11.955 Tf 11.955 0 Td[(n1n2+n2n1~a~b:Supposemiseven.Then~fmn1~fmn2=Fm )]TJ/F27 11.955 Tf 11.955 0 Td[()]TJ/F27 11.955 Tf 5.48 -9.684 Td[(n1+n2+m)]TJ/F25 7.97 Tf 6.587 0 Td[(1)]TJ/F27 11.955 Tf 11.955 0 Td[(n1+n2+m)]TJ/F25 7.97 Tf 6.587 0 Td[(1byLem.3.1.3=FmFn1+n2+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1byEqn..3.14.Nowsupposemisodd.Then~fmn1~fmn2=1 5)]TJ/F27 11.955 Tf 5.479 -9.683 Td[(Lmn1+n2+m)]TJ/F25 7.97 Tf 6.587 0 Td[(1+n1+n2+m)]TJ/F25 7.97 Tf 6.587 0 Td[(1)]TJ/F19 11.955 Tf 11.955 0 Td[(n1n2+n2n1=1 5LmLn1+n2+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1)]TJ/F19 11.955 Tf 11.955 0 Td[()]TJ/F19 11.955 Tf 9.298 0 Td[(1n1Ln2)]TJ/F28 7.97 Tf 6.586 0 Td[(n1byEqns..3.14,2.3.25: Corollary3.3.2 Forallpositiveintegersmandforallintegersn1andn2,~fmn2=FmF2n+m)]TJ/F25 7.97 Tf 6.587 0 Td[(1ifmiseven,1 5LmL2n+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1)]TJ/F19 11.955 Tf 11.955 0 Td[(2)]TJ/F19 11.955 Tf 9.298 0 Td[(1nifmisodd. .3.12 Proof.Observethat~fmn2=~fmn~fmn. 15

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Theorem3.3.3 Forallpositiveintegersmandforallintegersn1andn2,~`mn1~`mn2=5FmFn1+n2+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1ifmiseven,LmLn1+n2+m)]TJ/F25 7.97 Tf 6.587 0 Td[(1+)]TJ/F19 11.955 Tf 9.299 0 Td[(1n1Ln2)]TJ/F28 7.97 Tf 6.587 0 Td[(n1ifmisodd. .3.13 Proof.~`mn1~`mn2=n1~a+n1~bn2~a+n2~bbyEqn..2.9=n1+n2~a~a+n1+n2~b~b+n1n2+n2n1~a~b:Supposemiseven.Then~`mn1~`mn2=Fm)]TJ/F27 11.955 Tf 11.955 0 Td[()]TJ/F27 11.955 Tf 5.48 -9.684 Td[(n1+n2+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1)]TJ/F27 11.955 Tf 11.955 0 Td[(n1+n2+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1byLem.3.1.3=5Fm)]TJ/F27 11.955 Tf 5.479 -9.684 Td[(n1+n2+m)]TJ/F25 7.97 Tf 6.587 0 Td[(1)]TJ/F27 11.955 Tf 11.955 0 Td[(n1+n2+m)]TJ/F25 7.97 Tf 6.587 0 Td[(1=)]TJ/F27 11.955 Tf 11.955 0 Td[(byEqn..3.18=5FmFn1+n2+m)]TJ/F25 7.97 Tf 6.587 0 Td[(1byEqn..3.14:Nowsupposemisodd.Then~`mn1~`mn2=Lmn1+n2+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1+n1+n2+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1+n1n2+n2n1byLem.3.1.3=LmLn1+n2+m)]TJ/F25 7.97 Tf 6.587 0 Td[(1+)]TJ/F19 11.955 Tf 9.299 0 Td[(1n1Ln2)]TJ/F28 7.97 Tf 6.587 0 Td[(n1byEqn..3.15: Corollary3.3.4 Forallpositiveintegersmandforallintegersn,~`mn2=5FmF2n+m)]TJ/F25 7.97 Tf 6.587 0 Td[(1ifmiseven,LmL2n+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1+2)]TJ/F19 11.955 Tf 9.298 0 Td[(1nifmisodd. .3.14 Corollary3.3.5 Forallpositiveintegersmandforallintegersn1andn2,~`mn1~`mn2=5~fmn1~fmn2: .3.15 16

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Proof.Clearfrom.3.11and.3.13. Theorem3.3.6 Forallpositiveintegersmandforallintegersn1andn2,~fmn1~`mn2=5FmLn1+n2+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1ifmiseven,LmFn1+n2+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1+)]TJ/F19 11.955 Tf 9.299 0 Td[(1n1+1Fn2)]TJ/F28 7.97 Tf 6.586 0 Td[(n1ifmisodd. .3.16 Proof.~fmn1~`mn2=1 )]TJ/F27 11.955 Tf 11.955 0 Td[(n1~a)]TJ/F27 11.955 Tf 11.955 0 Td[(n1~bn2~a+n2~bbyThm.3.2.3=1 )]TJ/F27 11.955 Tf 11.955 0 Td[(n1+n2~a~a)]TJ/F27 11.955 Tf 11.955 0 Td[(n1+n2~b~b+n1n2)]TJ/F27 11.955 Tf 11.956 0 Td[(n2n1~a~b:Supposemiseven.Then~fmn1~`mn2=Fm)]TJ/F27 11.955 Tf 11.955 0 Td[()]TJ/F27 11.955 Tf 5.48 -9.684 Td[(n1+n2+m)]TJ/F25 7.97 Tf 6.587 0 Td[(1+n1+n2+m)]TJ/F25 7.97 Tf 6.587 0 Td[(1byLem.3.1.3=5FmLn1+n2+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1byEqns.2.3.15and.3.18:Nowsupposemisodd.Then~fmn1~`mn2=1 )]TJ/F27 11.955 Tf 11.955 0 Td[(Lm)]TJ/F27 11.955 Tf 5.479 -9.684 Td[(n1+n2+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1)]TJ/F27 11.955 Tf 11.955 0 Td[(n1+n2+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1+n1n2)]TJ/F27 11.955 Tf 11.955 0 Td[(n2n1byThm.3.2.3=LmFn1+n2+m)]TJ/F25 7.97 Tf 6.587 0 Td[(1+)]TJ/F19 11.955 Tf 9.299 0 Td[(1n1+1Fn2)]TJ/F28 7.97 Tf 6.587 0 Td[(n1byEqn..3.14: 3.4FibonacciandLucasidentitiesInthissectionwestateanumberofidentitiesinvolvingtheFibonacciandLucasnumberswhichfollowfromthedotproductformulasoftheprevioussection. Theorem3.4.1 Forallpositiveintegersmandforallintegersn1andn2,m)]TJ/F25 7.97 Tf 6.586 0 Td[(1Xj=0Fn1+jFn2+j=FmFn1+n2+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1ifmiseven,1 5LmLn1+n2+m)]TJ/F25 7.97 Tf 6.587 0 Td[(1)]TJ/F19 11.955 Tf 11.955 0 Td[()]TJ/F19 11.955 Tf 9.299 0 Td[(1n1Ln2)]TJ/F28 7.97 Tf 6.587 0 Td[(n1ifmisodd. .4.17 17

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Proof.Observethatbothsidesareequalto~fmn1~fmn2byTheorem3.3.1andthedenitionofdotproduct. Fromthisbasicidentityanumberofotheridentitiescanbederived.Westatetwoofthesimplestconsequencesnow. Corollary3.4.2 Forallpositiveintegersmandforallintegersn,mXj=0F2n+j=FmF2n+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1ifmiseven,1 5LmL2n+m)]TJ/F25 7.97 Tf 6.587 0 Td[(1)]TJ/F19 11.955 Tf 11.955 0 Td[(2)]TJ/F19 11.955 Tf 9.299 0 Td[(1nifmisodd.Proof.Taken1=n2inEqn.3.4.17. Corollary3.4.3 Forallpositiveintegersmandforallintegerskandn,Fn+m)]TJ/F25 7.97 Tf 6.586 0 Td[(2Fn)]TJ/F28 7.97 Tf 6.587 0 Td[(k+m)]TJ/F25 7.97 Tf 6.587 0 Td[(1+Fn)]TJ/F25 7.97 Tf 6.587 0 Td[(1Fn)]TJ/F28 7.97 Tf 6.587 0 Td[(k=Fm)]TJ/F25 7.97 Tf 6.587 0 Td[(1F2n)]TJ/F28 7.97 Tf 6.587 0 Td[(k+m)]TJ/F25 7.97 Tf 6.587 0 Td[(2ifmiseven,1 5)]TJ/F27 11.955 Tf 5.48 -9.684 Td[(Lm)]TJ/F25 7.97 Tf 6.586 0 Td[(1L2n)]TJ/F28 7.97 Tf 6.586 0 Td[(k+m)]TJ/F19 11.955 Tf 11.955 0 Td[(2)]TJ/F19 11.955 Tf 9.299 0 Td[(1n)]TJ/F28 7.97 Tf 6.587 0 Td[(kLk)]TJ/F25 7.97 Tf 6.587 0 Td[(1ifmisodd.Proof.Bythedenitionofdotproduct,theleft-handsideis~fmn)]TJ/F25 7.97 Tf 6.586 0 Td[(1~fmn)]TJ/F28 7.97 Tf 6.587 0 Td[(k)]TJ/F27 11.955 Tf 11.56 3.155 Td[(~fm)]TJ/F25 7.97 Tf 6.587 0 Td[(2n~fm)]TJ/F25 7.97 Tf 6.586 0 Td[(2n)]TJ/F28 7.97 Tf 6.587 0 Td[(k+1.Formeven,Theorem3.3.1gives~fmn)]TJ/F25 7.97 Tf 6.587 0 Td[(1~fmn)]TJ/F28 7.97 Tf 6.586 0 Td[(k)]TJ/F27 11.955 Tf 13.583 3.155 Td[(~fm)]TJ/F25 7.97 Tf 6.586 0 Td[(2n~fm)]TJ/F25 7.97 Tf 6.586 0 Td[(2n)]TJ/F28 7.97 Tf 6.586 0 Td[(k+1=FmF2n)]TJ/F28 7.97 Tf 6.587 0 Td[(k+m)]TJ/F25 7.97 Tf 6.586 0 Td[(2)]TJ/F27 11.955 Tf 11.955 0 Td[(Fm)]TJ/F25 7.97 Tf 6.586 0 Td[(2F2n)]TJ/F28 7.97 Tf 6.587 0 Td[(k+m)]TJ/F25 7.97 Tf 6.586 0 Td[(2=Fm)]TJ/F27 11.955 Tf 11.955 0 Td[(Fm)]TJ/F25 7.97 Tf 6.586 0 Td[(2F2n)]TJ/F28 7.97 Tf 6.586 0 Td[(k+m)]TJ/F25 7.97 Tf 6.587 0 Td[(2=Fm)]TJ/F25 7.97 Tf 6.587 0 Td[(1F2n)]TJ/F28 7.97 Tf 6.587 0 Td[(k+m)]TJ/F25 7.97 Tf 6.587 0 Td[(2:Formodd,Theorem3.3.1givesthat~fmn)]TJ/F25 7.97 Tf 6.587 0 Td[(1~fmn)]TJ/F28 7.97 Tf 6.587 0 Td[(k)]TJ/F27 11.955 Tf 13.584 3.155 Td[(~fm)]TJ/F25 7.97 Tf 6.587 0 Td[(2n~fm)]TJ/F25 7.97 Tf 6.587 0 Td[(2n)]TJ/F28 7.97 Tf 6.586 0 Td[(k+1equals1 5)]TJ/F27 11.955 Tf 5.479 -9.684 Td[(LmLn)]TJ/F25 7.97 Tf 6.587 0 Td[(1+n)]TJ/F28 7.97 Tf 6.586 0 Td[(k+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1)]TJ/F19 11.955 Tf 11.956 0 Td[()]TJ/F19 11.955 Tf 9.298 0 Td[(1n)]TJ/F25 7.97 Tf 6.586 0 Td[(1Ln)]TJ/F28 7.97 Tf 6.587 0 Td[(k)]TJ/F28 7.97 Tf 6.586 0 Td[(n+1)]TJ/F19 11.955 Tf 13.151 8.088 Td[(1 5Lm)]TJ/F25 7.97 Tf 6.587 0 Td[(2Ln+n)]TJ/F28 7.97 Tf 6.586 0 Td[(k+1+m)]TJ/F25 7.97 Tf 6.587 0 Td[(2)]TJ/F25 7.97 Tf 6.587 0 Td[(1)]TJ/F19 11.955 Tf 11.955 0 Td[()]TJ/F19 11.955 Tf 9.299 0 Td[(1nLn)]TJ/F28 7.97 Tf 6.586 0 Td[(k+1)]TJ/F28 7.97 Tf 6.586 0 Td[(n=1 5)]TJ/F27 11.955 Tf 5.479 -9.684 Td[(LmL2n)]TJ/F28 7.97 Tf 6.586 0 Td[(k+m)]TJ/F25 7.97 Tf 6.586 0 Td[(2)]TJ/F19 11.955 Tf 11.955 0 Td[()]TJ/F19 11.955 Tf 9.298 0 Td[(1n)]TJ/F25 7.97 Tf 6.586 0 Td[(1L)]TJ/F28 7.97 Tf 6.586 0 Td[(k+1)]TJ/F19 11.955 Tf 13.151 8.088 Td[(1 5Lm)]TJ/F25 7.97 Tf 6.587 0 Td[(2L2n)]TJ/F28 7.97 Tf 6.586 0 Td[(k+m)]TJ/F25 7.97 Tf 6.587 0 Td[(2)]TJ/F19 11.955 Tf 11.956 0 Td[()]TJ/F19 11.955 Tf 9.298 0 Td[(1nL)]TJ/F28 7.97 Tf 6.586 0 Td[(k+1=1 5LmL2n)]TJ/F28 7.97 Tf 6.586 0 Td[(k+m)]TJ/F25 7.97 Tf 6.586 0 Td[(2)]TJ/F19 11.955 Tf 13.151 8.088 Td[(1 5Lm)]TJ/F25 7.97 Tf 6.587 0 Td[(2L2n)]TJ/F28 7.97 Tf 6.587 0 Td[(k+m)]TJ/F25 7.97 Tf 6.587 0 Td[(2)]TJ/F19 11.955 Tf 13.15 8.088 Td[(1 5)]TJ/F19 11.955 Tf 9.299 0 Td[(1n)]TJ/F25 7.97 Tf 6.587 0 Td[(1L)]TJ/F28 7.97 Tf 6.586 0 Td[(k+1+1 5)]TJ/F19 11.955 Tf 9.298 0 Td[(1nL)]TJ/F28 7.97 Tf 6.586 0 Td[(k+1 18

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=1 5L2n)]TJ/F28 7.97 Tf 6.587 0 Td[(k+m)]TJ/F25 7.97 Tf 6.586 0 Td[(2Lm)]TJ/F27 11.955 Tf 11.955 0 Td[(Lm)]TJ/F25 7.97 Tf 6.587 0 Td[(2)]TJ/F19 11.955 Tf 13.151 8.088 Td[(1 5L)]TJ/F28 7.97 Tf 6.586 0 Td[(k+12)]TJ/F19 11.955 Tf 9.298 0 Td[(1n)]TJ/F25 7.97 Tf 6.587 0 Td[(1=1 5Lm)]TJ/F25 7.97 Tf 6.586 0 Td[(1L2n)]TJ/F28 7.97 Tf 6.587 0 Td[(k+m)]TJ/F25 7.97 Tf 6.587 0 Td[(2)]TJ/F19 11.955 Tf 11.955 0 Td[(2)]TJ/F19 11.955 Tf 9.299 0 Td[(1n)]TJ/F25 7.97 Tf 6.587 0 Td[(1L)]TJ/F28 7.97 Tf 6.586 0 Td[(k+1: Theideaofthepreviouscorollarycanbeusedtoderiveanumberofothermorecomplicatedidentitiesinvolvingmoresummandsofthesamesort. Theorem3.4.4 Forallintegersn1andn2andforallpositiveintegersm,mXj=0Ln1+jLn2+j=5FmFn1+n2+m)]TJ/F25 7.97 Tf 6.587 0 Td[(1ifmiseven,LmLn1+n2+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1+)]TJ/F19 11.955 Tf 9.298 0 Td[(1n1Ln2)]TJ/F28 7.97 Tf 6.586 0 Td[(n1ifmisodd. .4.18 Proof.Observethatbothsidesareequalto~fmn1~fmn2byTheorem3.3.3andthedenitionofdotproduct. Corollary3.4.5 Forallpositiveintegersmandforallintegersn,mXj=0L2n+j=5FmF2n+m)]TJ/F25 7.97 Tf 6.587 0 Td[(1ifmiseven,LmL2n+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1+2)]TJ/F19 11.955 Tf 9.298 0 Td[(1nifmisodd.Proof.Taken1=n2inEqn.3.4.18. Corollary3.4.6 Forallpositiveintegersmandforallintegerskandn,Ln+m)]TJ/F25 7.97 Tf 6.587 0 Td[(2Ln)]TJ/F28 7.97 Tf 6.586 0 Td[(k+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1+Ln)]TJ/F25 7.97 Tf 6.587 0 Td[(1Ln)]TJ/F28 7.97 Tf 6.586 0 Td[(k=5Lm)]TJ/F25 7.97 Tf 6.586 0 Td[(1L2n)]TJ/F28 7.97 Tf 6.587 0 Td[(k+m)]TJ/F25 7.97 Tf 6.587 0 Td[(2ifmiseven,Lm)]TJ/F25 7.97 Tf 6.586 0 Td[(1L2n)]TJ/F28 7.97 Tf 6.587 0 Td[(k+m)]TJ/F25 7.97 Tf 6.587 0 Td[(2+2)]TJ/F19 11.955 Tf 9.298 0 Td[(1n)]TJ/F25 7.97 Tf 6.586 0 Td[(1L)]TJ/F28 7.97 Tf 6.586 0 Td[(k+1ifmisodd.Proof.Bythedenitionofdotproduct,theleft-handsideis~`mn)]TJ/F25 7.97 Tf 6.586 0 Td[(1~`mn)]TJ/F28 7.97 Tf 6.586 0 Td[(k)]TJ/F27 11.955 Tf 10.256 3.155 Td[(~`m)]TJ/F25 7.97 Tf 6.586 0 Td[(2n~`m)]TJ/F25 7.97 Tf 6.586 0 Td[(2n)]TJ/F28 7.97 Tf 6.587 0 Td[(k+1.Formiseven,Theorem3.3.3gives~`mn)]TJ/F25 7.97 Tf 6.586 0 Td[(1~`mn)]TJ/F28 7.97 Tf 6.586 0 Td[(k)]TJ/F27 11.955 Tf 11.866 3.155 Td[(~`m)]TJ/F25 7.97 Tf 6.586 0 Td[(2n~`m)]TJ/F25 7.97 Tf 6.587 0 Td[(2n)]TJ/F28 7.97 Tf 6.586 0 Td[(k+1=5FmF2n)]TJ/F28 7.97 Tf 6.587 0 Td[(k+m)]TJ/F25 7.97 Tf 6.586 0 Td[(2)]TJ/F19 11.955 Tf 11.955 0 Td[(5Fm)]TJ/F25 7.97 Tf 6.587 0 Td[(2F2n)]TJ/F28 7.97 Tf 6.587 0 Td[(k+m)]TJ/F25 7.97 Tf 6.586 0 Td[(2 19

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=5Fm)]TJ/F27 11.955 Tf 11.955 0 Td[(Fm)]TJ/F25 7.97 Tf 6.586 0 Td[(2F2n)]TJ/F28 7.97 Tf 6.587 0 Td[(k+m)]TJ/F25 7.97 Tf 6.587 0 Td[(2=5Fm)]TJ/F25 7.97 Tf 6.587 0 Td[(1F2n)]TJ/F28 7.97 Tf 6.587 0 Td[(k+m)]TJ/F25 7.97 Tf 6.587 0 Td[(2:Formodd,Theorem3.3.3givesthat~`mn)]TJ/F25 7.97 Tf 6.587 0 Td[(1~`mn)]TJ/F28 7.97 Tf 6.586 0 Td[(k)]TJ/F27 11.955 Tf 11.865 3.155 Td[(~`m)]TJ/F25 7.97 Tf 6.587 0 Td[(2n~`m)]TJ/F25 7.97 Tf 6.586 0 Td[(2n)]TJ/F28 7.97 Tf 6.586 0 Td[(k+1equals)]TJ/F27 11.955 Tf 5.479 -9.684 Td[(LmLn)]TJ/F25 7.97 Tf 6.587 0 Td[(1+n)]TJ/F28 7.97 Tf 6.586 0 Td[(k+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1+)]TJ/F19 11.955 Tf 9.298 0 Td[(1n)]TJ/F25 7.97 Tf 6.586 0 Td[(1Ln)]TJ/F28 7.97 Tf 6.587 0 Td[(k)]TJ/F28 7.97 Tf 6.587 0 Td[(n+1)]TJ/F19 11.955 Tf 11.955 0 Td[(Lm)]TJ/F25 7.97 Tf 6.587 0 Td[(2Ln+n)]TJ/F28 7.97 Tf 6.587 0 Td[(k+1+m)]TJ/F25 7.97 Tf 6.587 0 Td[(2)]TJ/F25 7.97 Tf 6.586 0 Td[(1+)]TJ/F19 11.955 Tf 9.298 0 Td[(1nLn)]TJ/F28 7.97 Tf 6.586 0 Td[(k+1)]TJ/F28 7.97 Tf 6.586 0 Td[(n=)]TJ/F27 11.955 Tf 5.479 -9.683 Td[(LmL2n)]TJ/F28 7.97 Tf 6.587 0 Td[(k+m)]TJ/F25 7.97 Tf 6.586 0 Td[(2+)]TJ/F19 11.955 Tf 9.298 0 Td[(1n)]TJ/F25 7.97 Tf 6.586 0 Td[(1L)]TJ/F28 7.97 Tf 6.587 0 Td[(k+1)]TJ/F19 11.955 Tf 11.955 0 Td[(Lm)]TJ/F25 7.97 Tf 6.586 0 Td[(2L2n)]TJ/F28 7.97 Tf 6.587 0 Td[(k+m)]TJ/F25 7.97 Tf 6.586 0 Td[(2+)]TJ/F19 11.955 Tf 9.298 0 Td[(1nL)]TJ/F28 7.97 Tf 6.587 0 Td[(k+1=LmL2n)]TJ/F28 7.97 Tf 6.587 0 Td[(k+m)]TJ/F25 7.97 Tf 6.587 0 Td[(2)]TJ/F27 11.955 Tf 11.955 0 Td[(Lm)]TJ/F25 7.97 Tf 6.586 0 Td[(2L2n)]TJ/F28 7.97 Tf 6.586 0 Td[(k+m)]TJ/F25 7.97 Tf 6.586 0 Td[(2+)]TJ/F19 11.955 Tf 9.298 0 Td[(1n)]TJ/F25 7.97 Tf 6.586 0 Td[(1L)]TJ/F28 7.97 Tf 6.587 0 Td[(k+1)]TJ/F19 11.955 Tf 11.955 0 Td[()]TJ/F19 11.955 Tf 9.299 0 Td[(1nL)]TJ/F28 7.97 Tf 6.586 0 Td[(k+1=L2n)]TJ/F28 7.97 Tf 6.586 0 Td[(k+m)]TJ/F25 7.97 Tf 6.587 0 Td[(2Lm)]TJ/F27 11.955 Tf 11.955 0 Td[(Lm)]TJ/F25 7.97 Tf 6.586 0 Td[(2+2)]TJ/F19 11.955 Tf 9.299 0 Td[(1n)]TJ/F25 7.97 Tf 6.587 0 Td[(1L)]TJ/F28 7.97 Tf 6.586 0 Td[(k+1=Lm)]TJ/F25 7.97 Tf 6.587 0 Td[(1L2n)]TJ/F28 7.97 Tf 6.586 0 Td[(k+m)]TJ/F25 7.97 Tf 6.586 0 Td[(2+2)]TJ/F19 11.955 Tf 9.299 0 Td[(1n)]TJ/F25 7.97 Tf 6.587 0 Td[(1L)]TJ/F28 7.97 Tf 6.587 0 Td[(k+1: Theorem3.4.7 Forallintegersn1andn2andforallpositiveintegersm,mXj=0Fn1+jLn2+j=5FmLn1+n2+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1ifmiseven,LmFn1+n2+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1+)]TJ/F19 11.955 Tf 9.299 0 Td[(1n1+1Fn2)]TJ/F28 7.97 Tf 6.586 0 Td[(n1ifmisodd. .4.19 Proof.Observethatbothsidesareequalto~fmn1~`mn2byTheorem3.3.6andthedenitionofdotproduct. Corollary3.4.8 Forallpositiveintegersmandforallintegersn,mXj=0Fn+jLn+j=5FmL2n+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1ifmiseven,LmF2n+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1+)]TJ/F19 11.955 Tf 9.298 0 Td[(1n+1ifmisodd.Proof.Taken1=n2inEqn..4.19. 20

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Corollary3.4.9 Forallintegersk,nandforallpostiveintegersm,Fn+m)]TJ/F25 7.97 Tf 6.586 0 Td[(2Ln)]TJ/F28 7.97 Tf 6.587 0 Td[(k+m)]TJ/F25 7.97 Tf 6.587 0 Td[(1+Fn)]TJ/F25 7.97 Tf 6.587 0 Td[(1Ln)]TJ/F28 7.97 Tf 6.586 0 Td[(k=5Fm)]TJ/F25 7.97 Tf 6.587 0 Td[(1L2n)]TJ/F28 7.97 Tf 6.587 0 Td[(k+m)]TJ/F25 7.97 Tf 6.587 0 Td[(2ifmisevenLm)]TJ/F25 7.97 Tf 6.587 0 Td[(1F2n)]TJ/F28 7.97 Tf 6.587 0 Td[(k+m)]TJ/F25 7.97 Tf 6.587 0 Td[(2+2)]TJ/F19 11.955 Tf 9.299 0 Td[(1nF)]TJ/F28 7.97 Tf 6.587 0 Td[(k+1ifmisodd.Proof.Bythedenitionofdotproduct,theleft-handsideis~fmn)]TJ/F25 7.97 Tf 6.586 0 Td[(1~`mn)]TJ/F28 7.97 Tf 6.586 0 Td[(k)]TJ/F27 11.955 Tf 11.712 3.155 Td[(~fm)]TJ/F25 7.97 Tf 6.587 0 Td[(2n~`m)]TJ/F25 7.97 Tf 6.586 0 Td[(2n)]TJ/F28 7.97 Tf 6.587 0 Td[(k+1.Formeven,Theorem3.3.6gives~fmn)]TJ/F25 7.97 Tf 6.586 0 Td[(1~`mn)]TJ/F28 7.97 Tf 6.586 0 Td[(k)]TJ/F27 11.955 Tf 13.583 3.155 Td[(~fm)]TJ/F25 7.97 Tf 6.587 0 Td[(2n~`m)]TJ/F25 7.97 Tf 6.586 0 Td[(2n)]TJ/F28 7.97 Tf 6.587 0 Td[(k+1=5FmL2n)]TJ/F28 7.97 Tf 6.586 0 Td[(k+m)]TJ/F25 7.97 Tf 6.586 0 Td[(2)]TJ/F19 11.955 Tf 11.955 0 Td[(5Fm)]TJ/F25 7.97 Tf 6.586 0 Td[(2L2n)]TJ/F28 7.97 Tf 6.587 0 Td[(k+m)]TJ/F25 7.97 Tf 6.587 0 Td[(2=5Fm)]TJ/F27 11.955 Tf 11.955 0 Td[(Fm)]TJ/F25 7.97 Tf 6.587 0 Td[(2L2n)]TJ/F28 7.97 Tf 6.587 0 Td[(k+m)]TJ/F25 7.97 Tf 6.587 0 Td[(2=5Fm)]TJ/F25 7.97 Tf 6.587 0 Td[(1L2n)]TJ/F28 7.97 Tf 6.587 0 Td[(k+m)]TJ/F25 7.97 Tf 6.587 0 Td[(2:Formodd,Theorem3.3.6givesthat~fmn)]TJ/F25 7.97 Tf 6.587 0 Td[(1~`mn)]TJ/F28 7.97 Tf 6.587 0 Td[(k)]TJ/F27 11.955 Tf 13.584 3.155 Td[(~fm)]TJ/F25 7.97 Tf 6.587 0 Td[(2n~`m)]TJ/F25 7.97 Tf 6.587 0 Td[(2n)]TJ/F28 7.97 Tf 6.586 0 Td[(k+1equals)]TJ/F27 11.955 Tf 5.48 -9.684 Td[(LmFn)]TJ/F25 7.97 Tf 6.586 0 Td[(1+n)]TJ/F28 7.97 Tf 6.587 0 Td[(k+m)]TJ/F25 7.97 Tf 6.587 0 Td[(1+)]TJ/F19 11.955 Tf 9.299 0 Td[(1n)]TJ/F25 7.97 Tf 6.587 0 Td[(1+1Fn)]TJ/F28 7.97 Tf 6.587 0 Td[(k)]TJ/F28 7.97 Tf 6.587 0 Td[(n+1)]TJ/F32 11.955 Tf 11.955 9.683 Td[()]TJ/F27 11.955 Tf 5.48 -9.683 Td[(Lm)]TJ/F25 7.97 Tf 6.586 0 Td[(2Fn+n)]TJ/F28 7.97 Tf 6.587 0 Td[(k+1+m)]TJ/F25 7.97 Tf 6.586 0 Td[(2)]TJ/F25 7.97 Tf 6.586 0 Td[(1+)]TJ/F19 11.955 Tf 9.298 0 Td[(1n+1Fn)]TJ/F28 7.97 Tf 6.586 0 Td[(k+1)]TJ/F28 7.97 Tf 6.587 0 Td[(n=LmF2n)]TJ/F28 7.97 Tf 6.586 0 Td[(k+m)]TJ/F25 7.97 Tf 6.587 0 Td[(2+)]TJ/F19 11.955 Tf 9.299 0 Td[(1nF)]TJ/F28 7.97 Tf 6.587 0 Td[(k+1)]TJ/F32 11.955 Tf 11.955 9.684 Td[()]TJ/F27 11.955 Tf 5.48 -9.684 Td[(Lm)]TJ/F25 7.97 Tf 6.586 0 Td[(2F2n)]TJ/F28 7.97 Tf 6.586 0 Td[(k+m)]TJ/F25 7.97 Tf 6.586 0 Td[(2+)]TJ/F19 11.955 Tf 9.298 0 Td[(1n+1F)]TJ/F28 7.97 Tf 6.586 0 Td[(k+1=LmF2n)]TJ/F28 7.97 Tf 6.586 0 Td[(k+m)]TJ/F25 7.97 Tf 6.586 0 Td[(2)]TJ/F27 11.955 Tf 11.955 0 Td[(Lm)]TJ/F25 7.97 Tf 6.587 0 Td[(2F2n)]TJ/F28 7.97 Tf 6.587 0 Td[(k+m)]TJ/F25 7.97 Tf 6.586 0 Td[(2+)]TJ/F19 11.955 Tf 9.298 0 Td[(1nF)]TJ/F28 7.97 Tf 6.586 0 Td[(k+1)]TJ/F19 11.955 Tf 11.955 0 Td[()]TJ/F19 11.955 Tf 9.298 0 Td[(1n+1F)]TJ/F28 7.97 Tf 6.587 0 Td[(k+1=F2n)]TJ/F28 7.97 Tf 6.587 0 Td[(k+m)]TJ/F25 7.97 Tf 6.586 0 Td[(2Lm)]TJ/F27 11.955 Tf 11.955 0 Td[(Lm)]TJ/F25 7.97 Tf 6.587 0 Td[(2+2)]TJ/F19 11.955 Tf 9.298 0 Td[(1nF)]TJ/F28 7.97 Tf 6.586 0 Td[(k+1=Lm)]TJ/F25 7.97 Tf 6.587 0 Td[(1F2n)]TJ/F28 7.97 Tf 6.587 0 Td[(k+m)]TJ/F25 7.97 Tf 6.587 0 Td[(2+2)]TJ/F19 11.955 Tf 9.298 0 Td[(1nF)]TJ/F28 7.97 Tf 6.587 0 Td[(k+1: 21

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4AnglesinevendimensionInthischapterwestudytheanglesbetweenthevariousvectorswhichwestudiedsofar.Weshallrestrictourattentiontothecaseofevendimensionastheformulasbecomemuchmoreinvolvedinodddimension.WeshallcommentontheodddimensionalcaseinChapter5.Recallthattheanglebetweenarbitraryvectorsv1andv2satisescos=v1v2 kv1kkv2k: 4.1Dotproductswith~aand~bInthissection,wecomputethedotproductsofthevectors~aand~bwiththeFibonacciandLucasvectors. Lemma4.1.1 Forallintegersnandforallpositiveevenintegersm,~fmn~a=Fmn+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1; .1.1 ~fmn~b=Fmn+m)]TJ/F25 7.97 Tf 6.587 0 Td[(1: .1.2 Proof.~fmn~a=1 )]TJ/F27 11.955 Tf 11.955 0 Td[(n~a)]TJ/F27 11.955 Tf 11.955 0 Td[(n~b~abyThm.3.2.3=1 )]TJ/F27 11.955 Tf 11.955 0 Td[(n~a~a)]TJ/F27 11.955 Tf 11.955 0 Td[(n~b~a=1 )]TJ/F27 11.955 Tf 11.955 0 Td[(nFm)]TJ/F27 11.955 Tf 11.955 0 Td[(m)]TJ/F25 7.97 Tf 6.587 0 Td[(1byLem.3.1.3=Fmn+m)]TJ/F25 7.97 Tf 6.587 0 Td[(1: 22

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Lemma4.1.2 Forallintegersnandforallpositiveevenintegersm,~`mn~a=Fm)]TJ/F27 11.955 Tf 11.955 0 Td[(n+m)]TJ/F25 7.97 Tf 6.587 0 Td[(1; .1.3 ~`mn~b=Fm)]TJ/F27 11.955 Tf 11.955 0 Td[(n+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1: .1.4 Proof.Compute~`mn~a=n~a+n~b~abyThm.3.2.3=n~a~a+n~b~a=nFm)]TJ/F27 11.955 Tf 11.955 0 Td[(m)]TJ/F25 7.97 Tf 6.587 0 Td[(1byLem.3.1.3=Fm)]TJ/F27 11.955 Tf 11.955 0 Td[(n+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1:Equation.1.4isprovedsimilarly. 4.2Cosinesofangleswith~aand~bInlightofTheorem3.2.3,itisinterestingtoconsidertheanglesoftheFibonacciandLucasvectorswith~aand~b.Recallthatallthevectorsofinterestlieinthesameplane. Lemma4.2.1 Forallintegerskandallpositiveintegersm,~fm2kand~`m2k+1areontheoppositesideof~aas~b,and~fm2k+1and~`m2kareonthesamesideof~aas~b.Proof.Observethat<0,so2kispositiveand2k)]TJ/F25 7.97 Tf 6.586 0 Td[(1isnegative.TheresultfollowsfromTheorem3.2.3. Lemma4.2.2 Letn;mandn;mdenotetherespectiveanglesbetween~fmnand~aand 23

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between~fmnand~b.Thencosn;m=)]TJ/F27 11.955 Tf 11.955 0 Td[()]TJ/F25 7.97 Tf 6.586 0 Td[(1=2s 2n+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1 F2n+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1; .2.5 cosn;m=)]TJ/F27 11.955 Tf 11.955 0 Td[()]TJ/F25 7.97 Tf 6.586 0 Td[(1=2s )]TJ/F27 11.955 Tf 9.298 0 Td[(2n+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1 F2n+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1: .2.6 Proof.Computecosn;m=~fnm~a ~fnmk~ak=Fmn+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1 p FmF2n+m)]TJ/F25 7.97 Tf 6.587 0 Td[(1p Fm)]TJ/F27 11.955 Tf 11.956 0 Td[(m)]TJ/F25 7.97 Tf 6.587 0 Td[(1byLem.4.1.1,Cors.3.1.4,3.3.2=)]TJ/F27 11.955 Tf 11.955 0 Td[()]TJ/F25 7.97 Tf 6.586 0 Td[(1=2s 2n+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1 F2n+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1:Alsocosn;m=~fnm~b ~fnm~b=Fmn+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1 p FmF2n+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1p )]TJ/F27 11.955 Tf 9.298 0 Td[(Fm)]TJ/F27 11.955 Tf 11.955 0 Td[(m)]TJ/F25 7.97 Tf 6.587 0 Td[(1byLem.4.1.1,Cors.3.1.4,3.3.2=)]TJ/F27 11.955 Tf 11.955 0 Td[()]TJ/F25 7.97 Tf 6.586 0 Td[(1=2s )]TJ/F27 11.955 Tf 9.299 0 Td[(2n+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1 F2n+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1: Lemma4.2.3 Letn;mandn;mdenotetherespectiveanglesbetween~`mnand~aandbetween~`mnand~b.Thencosn;m=)]TJ/F27 11.955 Tf 11.956 0 Td[()]TJ/F25 7.97 Tf 6.587 0 Td[(1=2s 2n+m)]TJ/F25 7.97 Tf 6.587 0 Td[(1 F2n+m)]TJ/F25 7.97 Tf 6.587 0 Td[(1; .2.7 cosn;m=)]TJ/F27 11.955 Tf 11.956 0 Td[()]TJ/F25 7.97 Tf 6.587 0 Td[(1=2s 2n+m)]TJ/F25 7.97 Tf 6.587 0 Td[(1 F2n+m)]TJ/F25 7.97 Tf 6.587 0 Td[(1: .2.8 24

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Proof.Computecosn;m=~`nm~a ~`nmk~ak=Fm)]TJ/F27 11.955 Tf 11.955 0 Td[(n+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1 p 5FmF2n+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1p Fm)]TJ/F27 11.955 Tf 11.955 0 Td[(m)]TJ/F25 7.97 Tf 6.587 0 Td[(1byLem.4.1.1,Cors.3.1.4,3.3.2=)]TJ/F27 11.955 Tf 11.955 0 Td[()]TJ/F25 7.97 Tf 6.586 0 Td[(1=2s 2n+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1 F2n+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1:Equation.2.8isprovedsimilarly. Corollary4.2.4 Forallintegersnandforallevenintegersm,jn;mj>jn+1;mj.Proof.Observethatn;misaccutesincethecoecientof~aintheexpressionoffmnin3.2.9ispositive.Toshowthatjn;mj>jn+1;mj,itsucestoshowthatcosn;2kj+1=Fj+1,sotheresultfollowsfromEquation.2.5. Corollary4.2.5 Forallintegersnandforallpositiveintegerst,n;2t=n+t;2=)]TJ/F27 11.955 Tf 9.299 0 Td[(n+t=)]TJ/F27 11.955 Tf 9.299 0 Td[(n;2t.Inparticular,thesequencesofanglesfj;2tg1j=1,f)]TJ/F27 11.955 Tf 15.276 0 Td[(j;2tg1j=1,andft+j;2g1j=1areequal.Proof.Immediatefrom.2.5and.2.7. Sincethesequencesofanglesaresimplytailsofthoseindimensiontwo,weshalltakeanotherlookattheFibonaccivectorsoflength2intheSection4.5. 4.3AnglesbetweenvectorsInthissectionwestudytheanglesbetweenFibonacciandLucasvectors.Wexmtobeapositiveeveninteger. 25

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Lemma4.3.1 Forallintegersn1andn2,letn1;n2;mdenotetheanglebetween~fmn1and~fmn2.Thencosn1;n2;m=Fn1+n2+m)]TJ/F25 7.97 Tf 6.587 0 Td[(1 p F2n1+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1F2n2+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1:Proof.FollowsdirectlyfromTheorem3.3.1andCorollary3.3.2. Lemma4.3.2 Forallintegersn1andn2,letn1;n2;mdenotetheanglebetween~`mn1and~`mn2.Thencosn1;n2;m=Fn1+n2+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1 p F2n1+m)]TJ/F25 7.97 Tf 6.587 0 Td[(1F2n2+m)]TJ/F25 7.97 Tf 6.587 0 Td[(1:Proof.FollowsdirectlyfromTheorem3.3.3andCorollary3.3.4. Lemma4.3.3 Forallintegersn1andn2,letn1;n2;mdenotetheanglebetween~fmn1and~`mn2.Thencosn1;n2;m=5Ln1+n)]TJ/F25 7.97 Tf 6.587 0 Td[(2+m)]TJ/F25 7.97 Tf 6.587 0 Td[(1 p F2n1+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1F2n2+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1:Proof.FollowsdirectlyfromTheorem3.3.6andCorollaries3.3.2and3.3.4. Thevariousangleswhichwehavestudiedarenotindependent.Observethattheanglebetweenfmn1andfmn2canbecomputedbyeitheraddingorsubtractingtheirrespectiveangleswith~a.Ifn1andn2dierbyanevennumbertheyareonthesamesideof~asowesubtract,andifn1andn2dierbyanoddnumbertheyareonoppositesidesof~asoweadd.Thusforn1>n2,n1;n2;m=n1;m+)]TJ/F19 11.955 Tf 9.298 0 Td[(1n1)]TJ/F28 7.97 Tf 6.586 0 Td[(n2n2;m:Similarly,n1;n2;m=n1;m+)]TJ/F19 11.955 Tf 9.298 0 Td[(1n1)]TJ/F28 7.97 Tf 6.586 0 Td[(n2+1n2;m;n1;n2;m=n1;m+)]TJ/F19 11.955 Tf 9.298 0 Td[(1n1)]TJ/F28 7.97 Tf 6.586 0 Td[(n2n2;m: 26

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4.4AngleswiththeaxesLet~xidenotetheunitvectorinthedirectionofthei-thcoordinateaxisandletmbeapositiveeveninteger.Observethat~fmn~xi=Fn+i)]TJ/F25 7.97 Tf 6.587 0 Td[(1and~`mn~xi=Ln+i)]TJ/F25 7.97 Tf 6.586 0 Td[(1. Lemma4.4.1 Forallintegersn,let!m;n;ibetheanglebetween~fmnandthei-thstandardunitvectorthencos!m;n;i=Fn+i)]TJ/F25 7.97 Tf 6.586 0 Td[(1 p FmF2n+m)]TJ/F25 7.97 Tf 6.587 0 Td[(1: Lemma4.4.2 Forallintegersn,letm;n;ibetheanglebetween~lmnandthei-thstan-dardunitvectorthencosm;n;i=Ln+i)]TJ/F25 7.97 Tf 6.587 0 Td[(1 p 5FmF2n+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1: 4.5DimensiontwoInthissectionwediscusstheFibonacciandLucasvectorsoflengthtwo.ThisisdirectlyapplicabletoallevendimensionalcasesasdiscussedinSection4.2.WebeginwitharesultofLucaswhichgivesanicegeometricconditionontheFibonaccivectorsoflength2. Theorem4.5.1 LucasTheendpointsof~f2nlieontwohyperbolasgivenbytheequa-tiony2)]TJ/F27 11.955 Tf 11.955 0 Td[(yx)]TJ/F27 11.955 Tf 11.955 0 Td[(x2=1.InChapter6wewillconsiderthelimitofthesequenceofanglesoftheFibonacciandLucasvectorswith~a.Indimension2wecanalsoeasilyconsidertheangleswiththeaxesandthendeducetheangleswith~a. Lemma4.5.2 [4]Forallintegersk,limn!1Fn+k Fn=k: .5.9 27

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Lemma4.5.3 Forallintegersn,limn!1cos!2;n;x=1 1+2;limn!1cos!2;n;y= 1+2;limn!1cos2;n;x=1 1+2;limn!1cos2;n;y= 1+2:Proof.ByTheorem4.4.1cos!2;n;x=Fn p F2n+F2n+1=1 p 1+Fn+1=Fn2:ThusbyLemma4.5.2limn!1cos!2;n;x=1=+2:Theremaininglimitsareprovensimilarly. Itfollowsfromthepreviouslemmathatthelimitingunitvectorsforthedirectionsoff2nand`2nareboth1 1+224135=1 1+2~a:Inotherwordstheunitdirectionsapproachthatof~a.Weshallseethatthisisthecaseinalldimensions.Wenotethatindimensiontwo~aand~bformanorthogonalbasisforthewholevectorspace.FromTheorem3.2.3,weseethatthetransformationmatrixfrom~a,~btof2n1andf2n2is24n1)]TJ/F27 11.955 Tf 9.298 0 Td[(n1n2)]TJ/F27 11.955 Tf 9.298 0 Td[(n235: 28

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SincethismatrixisessentiallyVandermondeitisinvertible.Inotherwords,anytwodisinctFibonaccivectorsformabasisforR2.SimilarlyforLucasvectors. 29

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5CommentsonanglesinodddimensionAlthoughwedonotcarryoutthecomputationsofthecosinesofanglesinodddi-mensionbecausetheresultsarenotsonice,somecomputationsdoturnoutnicely.WepresenttheminthisChapter. 5.1Theanglebetween~aand~bRecallthat~aand~bformabasisfortheplanecontainingalloftheFibonacciandLucasvectors.Whenmiseven,~aand~bareorthogonal.Whenmisoddwehavethefollowing. Theorem5.1.1 Assumethatmisodd.Thenthecosineoftheanglebetween~aand~bis1=Lm.Proof.Letdenotetheanglebetween~aand~b.Thencos=~a~b k~ak~b=1 LmbyCor.3.1.4. Lemma5.1.2 Supposemisodd.Thenthevector)]TJ/F27 11.955 Tf 8.525 0 Td[(~a+Lmm)]TJ/F25 7.97 Tf 6.587 0 Td[(1~bisorthogonalto~aandonthesamesideof~aas~b. 30

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Proof.Compute~a)]TJ/F27 11.955 Tf 8.525 0 Td[(~a+Lmm)]TJ/F25 7.97 Tf 6.587 0 Td[(1~b=)]TJ/F27 11.955 Tf 8.525 0 Td[(~a~a+Lmm)]TJ/F25 7.97 Tf 6.586 0 Td[(1~a~b=)]TJ/F27 11.955 Tf 9.298 0 Td[(Lmm)]TJ/F25 7.97 Tf 6.586 0 Td[(1+Lmm)]TJ/F25 7.97 Tf 6.586 0 Td[(1byLem.3.1.3=0: 5.2Three-dimensionalFibonacciVectorsInthissectionwediscussthegeometricinterpretationsoftheseresultsin3dimensions.Thiscaseisnicerthanthegeneralodddimensionalcase. Example5.2.1 WiththenotationofLemma4.4.1,limn!1cos!3;n;x=1=;limn!1cos!3;n;y=1=2;limn!1cos!3;n;z==2:So,thelimitingunitvectoris1 2;1 2; 2=1 2h1;;2i.Thus,geometricallythelimitinglineisgivenby~rt=t;;2orx=t;y=t;andz=2t.Nowconsiderthedeterminantofthematrix~fn~fm~fr=FnFn+1Fn+2FmFm+1Fm+2FrFr+1Fr+2:Thelatterdeterminantequalszerosincethelastcolumnisthesumofthersttwocolumns.Therefore,thevectorsaredependentcoplanar.Weassumewithoutlossofgeneralityn
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z=x+y.Eachvector~fn=~fnliesintheintersectionofthisplanewiththeunitspherex2+y2+z2=1{thisintersectionisagreatcircleofthissphere. 32

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6AsymptoticsofanglesWeconsidertheultimatebehavioroftheanglesoftheFibonacciandLucasvectorswith~aand~b. 6.1Angleswith~a Lemma6.1.1 WithreferencetoLemma4.2.2,forallevenintegersm,limn!1cosn;m=1:ThatistosaythedirectionoftheFibonaccivectorsapproachesthatof~a.Proof.ByLemma4.2.5,cosn;m=)]TJ/F27 11.955 Tf 10.633 0 Td[()]TJ/F25 7.97 Tf 6.587 0 Td[(1=2p 2n+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1=F2n+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1ThusweconsiderFk=k.ByEquation.3.14,Fk=k=)]TJ/F27 11.955 Tf 9.999 0 Td[()]TJ/F25 7.97 Tf 6.586 0 Td[(1k)]TJ/F27 11.955 Tf 9.999 0 Td[(k=k=)]TJ/F27 11.955 Tf 9.999 0 Td[()]TJ/F25 7.97 Tf 6.587 0 Td[(1)]TJ/F19 11.955 Tf 9.999 0 Td[(=k.Observethatj=j<1and)]TJ/F27 11.955 Tf 11.955 0 Td[(.Thuslimk!1Fk=k=1 )]TJ/F27 11.955 Tf 11.955 0 Td[(:Itfollowsthatlimn!1cosn;m=1: Lemma6.1.2 WithreferencetoLemma4.2.3,forallevenintegersm,limn!1cosn;m=1: 33

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ThatistosaythedirectionoftheLucasvectorsapproachesthatof~a.Proof.SimilartothatofLemma6.1.1. 6.2DominanteigenvaluesInthelastsectionwesawthatinevendimension,thesequencesoftheFibonacciandLucasvectorsapproachindirectionthatofthevector~a.Wenowgiveanotherproofofthisfactwhichdoesnotdependupontheparityofthedimension.Weuseavariationofthewell-knownpowermethodtodoso. Denition6.2.1 SupposeisaneigenvalueofasquarematrixAthatislargerinabsolutevaluethananyothereigenvalueofA.TheniscalledthedominanteigenvalueofA.AneigenvectorcorrespondingtoiscalledadominanteigenvectorofA.Dominanteigenvaluesplayanimportantroleinthestudyofmatrices.Thepowermethodprovidesameansofndingthedominanteigenvalueandeigenvector.Westateoneofthemoregeneralformsofthepowermethod. Theorem6.2.2 AssumethatthemmcomplexmatrixAhasadominanteigenvalueandauniquedominanteigenvector~vuptoscalarmultiples.ThenthesequenceA~x=kA~xk,A2~x=kA2~xk,A3~x=kA3~xk,...,convergesto~v=k~vkforanyintitialvector~xexceptforasetofmeasurezero.Intheliteratureonendsvariouscriteriaontheinitialvectorfortheconvergenceoftheabovesequencetoadominanteigenvector.Clearly,anyvectorinthesumofeigenspacesotherthanthatofthedominanteigenvaluecannotconvergeadominanteigenvalue.Foradiagonalizablematrix,thisisalmostasucientcondition. Theorem6.2.3 AssumethatthemmcomplexmatrixAisdiagonalizableandhasadominanteigenvalueandauniquedominanteigenvector~vuptoscalarmultiples.Let~v1=~v,~v2,...,~vmdenoteabasisofeigenvectorsforthecomplexvectorspaceof 34

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columnsvectorswithcomplexentriesandlengthm.Let~x=c1v1+c2v2++cmvmwithc16=0.ThenthesequenceA~x=kA~xk,A2~x=kA2~xk,A3~x=kA3~xk,...convergesto~v=k~vk.Proof.ComputeA~x=Ac1~v1+c2~v2++cm~vm=c1A~v1+c2A~v2++cmA~vm=c11~v1+c22~v2++cmm~vm;whereiistheeigenvalueassociatedwith~viforalli.ByrepeatedmultiplicationbyA,Ak~x=c1k1~v1+c2k2~v2++cmkm~vm:ThusAk~x=k"c1~v1+c22 k~v2++crr k~vr#:Butisadominanteigenvalue,soitislargerinabsolutevaluethanallothereigen-values,sothatji=j<1fori=2,3,...,r.Thuseachfractioni=kapproaches0askgoestoinnity. Letusmodifytheprevioustheoremslightlytomakeitmoreapplicabletooursituation. Theorem6.2.4 AssumethatthemmcomplexmatrixAasadominanteigenvalueanddominanteigenvector~v.Let~v1=~v,~v2,...,~vrdenoteamaximalsetoflinearlyindependentnonzeroeigenvectorsandlet~vr+1,~vr+2,...,~vsdenoteabasisofgeneralizedeigenvectorsforthegeneralizedeigenspaceassociatedwithzero.Let~x=c1v1+c2v2++crvrwithc16=0.ThenthesequenceA~x=kA~xk,A2~x=kA2~xk,A3~x=kA3~xk,...convergesto~v=k~vk. 35

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Proof.Observethatforsomej,Aj~viisaneigenvectorofAforalliis.NowapplyTheorem6.2.3withArestrictedtotheeigenspacesand~y=Aj~xinplaceof~x.NowtheresultfollowssinceAk~y=Ak+j~x. ObservethatthematrixTofEqn..1.1haseigenvalues0,,withrespectivealgebraicmultiplicitesm)]TJ/F19 11.955 Tf 13.194 0 Td[(2,1,1andrespectivegeometricmultiplicites1,1,1.Thusthecomplexvectorspaceofcomplexvectorsoflengthmhasabasisconsistingofnonzeroeigenvectorsandgeneralizedeigenvectorsassociatedwith0.Abasisofthegeneralizedeigenspaceassociatedwith0hasabasisofgeneralizedeigenvectorsconsistingofe1=266666666666410...0003777777777775;e2=266666666666401...0003777777777775;;em)]TJ/F25 7.97 Tf 6.586 0 Td[(2=266666666666400...1003777777777775:Thusthecolumnvectorspacehasabasis~a,~b,e1,e2,...,em)]TJ/F25 7.97 Tf 6.586 0 Td[(2.Thisgivesusthefollowing. Corollary6.2.5 LetTbeasinEqn..1.1.Let~x=ca~a+cv~b+c1e1+cm)]TJ/F25 7.97 Tf 6.587 0 Td[(2em)]TJ/F25 7.97 Tf 6.587 0 Td[(2withc6=0.ThenthesequenceT~x=kT~xk,T2~x=kT2~xk,T3~x=kT3~xk,...convergesto~a=k~ak. Corollary6.2.6 ThesequencesofdirectionsoftheFibonacciandLucasvectorsap-proachthatofthevector~aProof.ImmediatefromLemma3.2.2andCorollary6.2.5. 6.3FurtherdirectionsWehavebeguntoconsideranumberofvariationsontheworkpresentedinthisthesis. 36

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ConsiderthereverseFibonacciandLucasvectors~rmn=26666664Fn+m)]TJ/F25 7.97 Tf 6.587 0 Td[(1Fn+m)]TJ/F25 7.97 Tf 6.587 0 Td[(2...Fn37777775;~tmn=26666664Ln+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1Ln+m)]TJ/F25 7.97 Tf 6.586 0 Td[(2...Ln37777775:Identicalresultsholdforthesevectorsasdidfor~fmn,~`mn.However,anumberofveryinterestingidentitiesarisewhenweconsider~fmn~rmn,~fmn~tmn,~`mn~rmn,and~`mn~tmn. ConsidertheFibonacciandLucasvectorswithastepofp:~fm;pn=26666664FnFn+1p...Fn+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1p37777775;~`m;pn=26666664LnLn+1p...Ln+m)]TJ/F25 7.97 Tf 6.586 0 Td[(1p37777775:Ourcomputationsstillworkoutsincethisconstantstepgivesrisetogeometricseriesinandasbefore.Again,interestingidentitiesarisefromthecompu-tationsofdotproducts.Dierentstepscanbemixed,aswellasreversevectorsconsidered. Tosomeextentourcomputationscanbecarriedoutforanysecondorderre-currencexn+2=cxn+dxn+1.However,whenc6=1so6=)]TJ/F19 11.955 Tf 9.298 0 Td[(1manynicepropertiesvanish.Anumberofotherproblemssuggestthemselves ConsidertheKroneckerproductsoftheFibonaccivectors.Weexpectthatsuchconsiderationswillgiverisetoidentitiesinvolvingsumsoflongerproducts. ReduceFibonacciandLucasvectorsmodkforanymodulusk.Thesevectorscyclethroughanitecollectionratherthanapproachagivenline.Thereare 37

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manydeepopenproblemsconcerningtheFibonaccinumbersmodulok.Perhapswemaygainsomeinsightfromaconsiderationofthesevectors. 38

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References [1] R.A.Askey,Fibonacciandrelatedsequences,MathematicsTeacher97004,116{119. [2] R.A.Askey,FibonacciandLucasnumbers,MathematicsTeacher98005,610{615. [3] V.E.Hoggatt,FibonacciandLucasNumbers",HoughtonMiinCompany,NewYork,NY,1969. [4] R.Honsberger,MathematicalGemsIII",Math.Assoc.America,Washington,DC,1985. [5] D.Huang,Fibonacciidentitiesmatrices,andgraphs",MathematicsTeacher98005,400{403. [6] V.J.Katz,AHistoryOfMathematics,AnIntroduction",2nded.,Addison-Wesley,Reading,MA,1998. [7] R.E.LarsonandB.H.Edwards,ElementaryLinearAlgebra",D.C.HeathandCompany,Lexington,MA,1969. [8] L.Sadum,AppliedLinearAlgebra:thedecouplingprinciple",Prentice{Hall,UpperSaddleRiver,NJ,2001. [9] G.Strang,IntroductiontoLinearAlgebra",2nded.,Wellesley-CambridgePress,Wellesley,MA,1998. 39

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AbouttheAuthorGrowingupIwouldhaveneverdreamedthatIwouldwriteathesisinmathematics.Afterall,theonesubjectIfearedthemostwasmathematics.ThenonedayitallchangedwhileIwasincollegeatGeorgiaSouthernUniversity.ItwastherethatImetpeoplewhowerewillingtotakethetimeouttoexplainthemanythingsthatconfusedmeanditwastherethatIdecidedIwantedtobecomeamathprofessor.Formanyyearsmyfearofmathematicsmademeshyawayfromlearningaboutseveralothersubjects.Itismygoaltomakesurethatnostudentofminewilleverletthesubjectofmathematicsgetinthewayofthedreamstheywishtopursue.IhopethatinmylifetimeIcanbehalftheteacherthatmyprofessorsweretome.


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Fibonacci vectors
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ABSTRACT: By the n-th Fibonacci (respectively Lucas) vector of length m, we mean the vector whose components are the n-th through (n+m-1)-st Fibonacci (respectively Lucas) numbers. For arbitrary m, we express the dot product of any two Fibonacci vectors, any two Lucas vectors, and any Fibonacci vector and any Lucas vector in terms of the Fibonacci and Lucas numbers. We use these formulas to deduce a number of identities involving the Fibonacci and Lucas numbers.
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