Joy Visualization and Graphics Research Group Department of Computer Science University of California Davis These notes give the de64257nition of a vector space and several of the concepts related to these spaces Examples are drawn from the vector s ID: 33127
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Associativity–foranythreevectors~v1,~v2and~v3inV,(~v1+~v2)+~v3=~v1+(~v2+~v3)Thisruleisillustratedinthegurebelow.Onecanseethateventhoughthesum~v1+~v2+~v3iscalculateddifferently,theresultisthesame. ZeroVector–thereisauniquevectorinVcalledthezerovectoranddenoted~0suchthatforeveryvector~v2V~v+~0=~0+~v=~vAdditiveInverse–foreachelement~v2V,thereisauniqueelementinV,usuallydenoted�~v,sothat~v+(�~v)=~0Inthecaseof2-dimensionalvectors,�~vissimplyrepresentedasthevectorofequalmagnitudeto~v,butintheoppositedirection. Theuseofanadditiveinverseallowsustodeneasubtractionoperationonvectors.Simply~v2�~v1=~v2+(�~v1)Theresultofvectorsubtractioninthespaceof2-dimensionalvectorsisshownbelow.2 Frequentlythis2-dvectorsisprotrayedasjoiningtheendsofthetwooriginalvectors.Aswecansee,sincethevectorsaredeterminedbydirectionandlength,andnotposition,thetwovectorsareequivalent. ScalarMultiplicationassociateswitheveryvector~v2Vandeveryscalarc,anotheruniquevector(usuallywrittenc~v), Forscalarmultiplicationthefollowingpropertieshold:Distributivity–foreveryscalarcandvectors~v1and~v2inV,c(~v1+~v2)=c~v1+c~v2Inthecaseof2-dimensionalvectors,thiscanbeeasilyseenbyextendingtheparallelogramillustrationabove.Wecanseebelowthatthesumofthevectors2~v1and2~v2isjusttwicethevector~v1+~v23 DistributivityofScalars–foreverytwoscalarsc1andc2andvector~v2V,(c1+c2)~v=c1~v+c2~vAssociativity–foreverytwoscalarsc1andc2andvector~v2V,c1(c2~v)=(c1c2)~vIdentity–foreveryvector~v2V,1~v=~v2ExamplesofVectorSpacesExamplesofvectorspaceaboundinmathematics.Themostobviousexamplesaretheusualvectorsin2,fromwhichwehavedrawnourillustrationsinthesectionsabove.Butwefrequentlyutilizeseveralothervectorsspaces:The3-dspaceofvectors,thevectorspaceofallpolynomialsofaxeddegree,andvectorspacesofnnmatrices.Webrieydiscussthesebelow. TheVectorSpaceof3-DimensionalVectorsThevectorsin3alsoformavectorspace,whereinthiscasethevectoroperationsofadditionandscalarmultiplicationaredonecomponentwise.Thatis~v1=x1;y1;z1&#x-433;and~v2=x2;y2;z2&#x-433;arevectors,thenadditionis~v1+~v2=x1+x2;y1+y2;z1+z2&#x-278;4 and,ifcisascalar,scalarmultiplicationisgivenbyc~v1=cx1;cy1;cz1&#x-278;Theaxiomsareeasilyveried(forexampletheadditiveidentityof~v1=x1;y1;z1&#x-295;isjust�~v1=�x1;�y1;�z1&#x]TJ/;༔ ;.9; T; -4;Y.5; -;.0; Td;&#x[000;,andthezerovectorisjust~0=0;0;0&#x]TJ/;ø 1;�.90; Tf;&#x 12.;Է ;� Td;&#x[000;.Heretheaxiomsjuststatewhatwealwayshavebeentaughtaboutthesesetsofvectors. VectorSpacesofPolynomialsThesetofquadraticpolynomialsoftheformP(x)=ax2+bx+calsoformavectorspace.Weaddtwoofpolynomialsbyaddingtheirrespectivecoefcients.Thatis,ifp1(x)=a1x2+b1x+c1andp2(x)=a2x2+b2x+c2,then(p1+p2)(x)=(a1+a2)x2+(b1+b2)x+(c1+c2)andmultiplicationisdonebymultiplyingthescalarbyeachcoefcient.Thatis,ifsisascalar,thensp(x)=(sa)x2+(sb)x+(sc)Theaxiomsareagaineasilyveriedbyperformingtheoperationsindividuallyonliketerms. Asimpleextensionoftheaboveistoconsiderthesetofpolynomialsofdegreelessthanorequalton.Itiseasilyseenthatthesealsoformavectorspace. VectorSpacesofMatricesThesetofnnMatricesformavectorspace.Twomatricescanbeaddedcomponentwise,andamatrixcanbemultipliedbyascalar.Allaxiomsareeasilyveried.5 Assumingthatc1isnotzero,wecanseethat~v1=c2 c1~v2++cn c1~vnAnysetofvectorscontainingthezerovector(~0)islinearlydependent.3.2.1ExampleTogiveanexampleofalinearindependentsetthateveryonehasseen,considerthethreevectors~i=1;0;0&#x]TJ/;ø 1;�.90; Tf;&#x 11.;ԕ ;� Td;&#x[000;;~j=0;1;0&#x]TJ/;ø 1;�.90; Tf;&#x 11.;ԕ ;� Td;&#x[000;;~k=0;0;1&#x]TJ/;ø 1;�.90; Tf;&#x 11.;ԕ ;� Td;&#x[000;inthevectorspaceofvectorsin3Considertheequationc1~i+c2~j+c3~k=~0Ifwesimplifyleft-handsidebyperformingtheoperationscomponentwiseandwritetheright-handsidecomponentwise,wehavec1;c2;c3&#x-278;=0;0;0&#x]TJ/;ø 1;�.90; Tf;&#x 11.;ԕ ;� Td;&#x[000;whichcanonlybesolvedifc1=c2=c3=0.3.3ABasisforaVectorSpaceLet~v1;~v2;:::;~vnbeasetofvectorsinavectorspaceVandletSbethespanofV.If~v1;~v2;:::;~vnislinearlyindependent,thenwesaythatthesevectorsformabasisforSandShasdimensionn.SincethesevectorsspanS,anyvector~v2Scanbewrittenuniquelyas~v=c1~v1+c2~v2++cn~vn7