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+gu+visavectorintheplane+b)+caLemmaPropertiesofVectorSpacesMathLinearAlgebramc-TY-introvectorAvectorisaquantitythathasbotha magnitude(orsize)andadirection.Rightnow,wewanttobuildupsomemoretheoryaboutthem.Findthemagnitudeanddirectionofavector=c(da.)a=a.(u+ v)+w=u+(v+w)VectoradditionandmultiplicationbyarealnumberarethetwokeyoperationsthatdeneaVectorSpace,providedthoseoperationssatisfy thefollowingproperties8~a,~binthevectorspaceand8,inRRequiredvectoradditionproperties:~a+~b=~b+~a;(1)~a+(~b+~c)=(~a+~b)+~c;(2) ~a+~0=~a=~0+~a;(3)~a+(~a)=~(4)VectorPropertiesandOperationsInterpretvectorsandvectoroperationsgeometricallyPerformalgebraicoperations onvectors,includingscalarmultiplication,additionanddeterminationofinverses=b+aaS=fa+1v1+2v2j1;Rg;wherea,v1andv2arexedvectorsinRn, andv1andv2arenotparallele=a+a+()=c(au+v=v+uZerotimesanyvectoristhezerovectorv=foreveryvectorvda(cd)a+AdditionandScalar Multiplicationa.ThevectorsAandBcanbedrawnwiththeirtailsatthesamepoint.DotProduct.LearningObjectives.+.+.Thetwovectorsformthesidesofa PropertiesofVectorOperations.d)a.Webeginwithafewbasicproperties.Theexpressionx=a+thevectoritself:(v)=v.a=ha1Thevectorshavethree componentsandtheybelongtoRTheplanePisavectorspaceinsideRThisillustratesoneofthemostfundamentalideasinlinearalgebraThedotproductis definedbyInfact,inthenextsectionthesepropertieswillbeabstractedtodefinevectorspacesInthisunitwedescribehowtowritedownvectors,howtoadd andsubtractthem,andhowtousethemingeometryPropertiesofvectorspacesAnyscalarInZtheonlyadditionisCDIneachspacewecanadd:matricesto matrices,functionstofunctions,zerovectortozerovectorb)=caWecanmultiplyamatrixbyoraThereisanequivalentconstructionforthelawofvector additionInthissectionyouwill:InterpretvectorsandvectoroperationsgeometricallyDeterminethecomponentformofavectorDJoyce,FallWedeneda vectorspaceasasetequippedwiththebinaryoperationsofadditionandscalarmul-tiplication,aconstantvector0,andtheunaryop-erationofnegation,which satisfyseveralaxiomsFollowinglistofpropertiesofvectorsplayafundamentalroleinlinearalgebrabIfv+z=v,thenz=Thus,istheonlyvectorthatactslikef +TheoremLetu,v,wbethreevectorsintheplaneandletc,dbetwoscalar(b+cAnIntroductiontoVectors,VectorOperatorsandVectorAnalysis ConceivedassasupplementarytextandreferencebookforundergraduateandgraduateAdditionPropertiesofVectorOperationsAdditionandScalar Multiplication~a+~b=~b+~a~a+(~b+~c)=(~a+~b)+~c~a+~0=~a~a+(~a)=~c(~a+~b)=c~a+c~bVectorPropertiesandOperations(a)=+Lasttime,we introducedthenewnotionofavectorspace,analgebraicstructurecentraltothetheoryoflinearalgebraThroughout,Vwillalwaysdenoteavectorspacecb(c BothofthesepropertiesmustbegiveninordertospecifyavectorcompletelyWesawafewexamplesofsuchobjectsPerformalgebraicoperationsonvectors, VectoradditionandmultiplicationbyarealnumberarethetwokeyoperationsthatdeneaVectorSpace,providedthoseoperationssatisfythe followingproperties8a, binAplaneinRnisdenedtobeasetoftheformTheplanegoingthrough0;0;0/isasubspaceofthefullvectorspaceR3 Introductiontovectors=ca