Solution Manual For Introduction To Linear Algebra, 5th Edition

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MarieVaniskoDouglasB.MeadeIntroductiontoLinearAlgebraaFIFTHEDITION:LeeW.Iohnson®R.DeonRipsmJimmyI.fmoldSoCeeelaEeEe

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DownloadedfromStudyXY.com®+StudyXYSdYe.o>\|iF’prE\3SStudyAnythingThisContentHasbeenPostedOnStudyXY.comassupplementarylearningmaterial.StudyXYdoesnotendroseanyuniversity,collegeorpublisher.Allmaterialspostedareundertheliabilityofthecontributors.wv8)www.studyxy.com

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BASICOPERATIONS1_—MapleTechnologyResourceManualMapleisapopularandpowerfulsoftwaretoolusedforthenumeric,symbolic,andgraphicanalysisofmathematicalproblems.Unlikemosttraditionalcomputationalsoftware,acomputeralgebrasystem(CAS)suchasMapleisdesignedtoperformexactsymbolicmanipulationsofmathematicalobjects,includingvectorsandmatrices.ThepurposeofthissupplementistointroducethebasicMaplecommandsneededtostartusingMapleasatoolforexploringthelinearalgebratopicsdevelopedinthistext.Maple,likeanyothercomputeralgebrasystem(CAS),isdesignedtoperformsymboliccomputations.Thatis,Maplecanmanipulateandprocessmathematicalobjectssuchasexpressions,equations,sets,vectors,andmatricesusingtherulesofalgebratherebyavoidingmanyofthenumericalissuesthatareinvolvedinmostfloating-pointnumericcomputations.Maplecanalsoperformnumericcomputations,butthefocusofthissupplementisontheexactsymboliccomputationsrelatedtolinearalgebra.1BasicOperationsLaunchMaple.WhenMapleopensalargewindowwillappear.Thiswindowprovidesanenvironmentforcreating,modifying,andviewingMapledocuments,calledworksheetsor.mwsfiles.AMapleworksheetcancontaintext,input,output,andgraphics(includinganimations).Worksheetsareplatform-independentASCIIfilesthatareeasilytransferredbetweenusersviae-mailortheWWW.Maplecanbeusedforsimplecomputations,suchasthefollowing:>8*17+49;185>alpha:=sin(Pi/3);1a=3V3If,insteadofasemi-colon,acolonisusedtoterminateaMaplecommand,theoutputofthecommandissuppressed:>beta:=sqrt(1+7/10*alpha):Ifneitherasemi-colonnoracolonisusedtoterminateaMaplecommand,anerrormessagewillbegenerated:>beta”2Warning,prematureendofinput‘Whenthisoccurs,itisnotnecessarytore-typethecommand.Simplypositionthecursorattheendofthecommand,typetheappropriateterminalcharacterandre-executethecommand.Notethatassignmentsaremadeusing“:="and“=”isusedtoconstructequations:>beta=evalf(beta);175V100+353=1267366475Maplehasanextensiveonlinehelpdatabase.Typehelp(topic);ortheshorthand?topictogethelponaspecifictopic.ForafirstlookatMaple’slinearalgebracapabilities,enterandexecutethecommand?linalg.TheHelppull-downmenuinthemenubarofMaple’sgraphicaluserinterfaceprovidessearchtoolsforaccessingMaple’shelpdocumentswhenthespecifictopicorMaplecommandnameisnotknown.—StudyXY

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2MAPLETECHNOLOGYRESOURCEMANUALInordertohaveameansofaccessingpreviousresultsitisagoodideatoassignresultstounique(andmeaningful)names.Maplenamesarecase-sensitive(e.g.,theconstant7isPi,notpinorPI).See?namesforadditionalinformationaboutvalidnamesinMaple.Exercisesfor§12.Evaluatetan14.32.Hint:See?mathforexamplesshowingtheuse3.Evaluate([sin41)*usingexactofsomeofMaple’sbasicmathematicalarithmetic,using32-,16-,8-,4,2-,andfunctions.1-digitfloating-pointarithmetic.(Note1.Evaluate3.132+|cosv/7|.thattheresultsarenotalwaysthesameasroundingthisnumbertothespecifiednumberofdigits.)2EnteringMatricesSupposewewishtoenterthematrix1-10—11A=|131].0ee1Priortoenteringthematrixitisnecessarytousethecommandwith(linalg);toloadMaple’slinalgpackage.ToenterAtypethecommand:>A:=matrix(3,3,[1,-1,0,1/2,3,1/3,0,exp(Pi),1]);1-1011A=|3330e™1Thefirsttwoargumentsofthismatrixcommandspecifythenumberofrowsandcolumnsinthematrix.Ifthematrixdimensionserenotstatedexplicitly,extrasquarebrackets(“[”,|”)mustbeusedtoseparatetherows:>Ac:=matrix([[1,-1,03,[1/2,3,1./31,[0,exp(Pi),111);1-101A=53.33333333330em1See?matrixforafulldescriptionofthesyntaxofthematrixcommand.Noticethatthe(2,3)entryofAhasbeenconvertedtoafloating-pointapproximation,butnottheotherfractionortheexponentialfunction.Ingeneral,Maplepreservesexactvaluesuntilafloating-pointnumberappearsinanexpressionoruntiltheevalfcommandisused.Afloating-pointversionofthismatrix,withfive(5)significantdigits,isobtainedwith:>evalf(evalm(A),5);1.-10..500003..333330.23.1411.ThedefaultnumberofsignificantdigitsiscontainedintheMaplenameDigits.Unlessexplicitlychanged,allfloating-pointcomputationsareperformedwithten(10)significantdigits.Thatis,atthestartofaMaplesession,theassignmentDigits:=10:isexecuted.—StudyXY

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THEGAUSSELIMANDGAUSSJORDCOMMANDS3Therowandcolumnindicescanbeusedtoaccessindividualelementsofamatrix.Forexample,tonegatethe(2,1)elementofA,type:>A[2,1]:=-A[2,1];-1Agi=5Toseethecurrentvalueofamatrix,usetheevalmcommand:>evalm(A);1-10-153.33333333330e"1Exercisesfor§22.Obtainthefloating-pointversionofthe1.CreatethematrixQbychangingthematrixQcreatedinExercise1(1,3)entryofAfrom0to2.3ThegausselimandgaussjordCommandsTheMaplecommandsgausselimandgaussjorduseelementaryrowoperationstotransformamatrixtoechelonformandreducedechelonform,respectively.Toillustrate,considertheproblemofsolvingthesystemzy+229+223+x4=32z7+4p+3x3+44=6ThepairoflinearequationscanbeenteredinMapleas:>eql:= x[1]+2xx[2]+2%x[3]+x[4]=3:>eq2:=2xx[1]+4xx[2]+3*x[3]+4*x[4]=6:Theaugmentedcoefficientmatrixis:>A:=genmatrix([eql,eq2],[x[i]$i=1..4],flag);12213A=|24346Thereducedechelonformoftheaugmentedmatrixis:>Ar:=gaussjord(A);12053a=}01-2HThus,theoriginalsystemisequivalenttoBx=bwith:>B:=submatrix(Ar,1..2,1..4);>b:=convert(submatrix(Ar,1..2,[5]),vector);[tr205B=E01—2b:=[3,0]—StudyXY

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4MAPLETECHNOLOGYRESOURCEMANUALThetwolinearequationsinthereducedsystemare:>sol:=geneqns(B,x,b);sol:={z1+222+524=3,23—24=0}Fromwhichthesolutionsareseentobe:>solve(sol,{x[11,x[3]1});{#1=—229—524+3,23=224}ThelinsolvecommandreturnsthegeneralsolutionwhenthesystemhasaninfinitenumberofsolutionsandNULL(theemptyresponse)whenthesystemofequationsisinconsistent.Exercisesfor§32.SolvethesysteminExercise31(Section1.SolvethesysteminExercise30(Section1.2).1.2).4SurgeryonMatricesMapleprovidesconvenienttoolsforthemanipulationofindividualelementsandsubmatrices(blocks)ofexistingmatricestocreatenewmatrices.Toillustrate,considerthe3x4matrix:>B:=matrix(3,4,[[1,3,-5,0],>[9,6,-2,11,>[o,4,4,111):ToextractanindividualelementofB,sayby3,use:>b23:=B[2,3];b23:=—2Toextractthe2x2blockfromtheupper-rightcornerofB,type:>Bl:=submatrix(B,[1,2],[3,4]);-50NET)ThiscommandinstructsMapletodefineanewmatrix,B1,madefromrows1and2ofBandcolumns3and4ofB.Anequivalentwaytoachievethesameresultis:>Bl:=submatrix(B,1..2,3..4);-50NET]Therangenotationlo..hiisanabbreviationforallintegersbeginningwithloandendingwithhi.Similarly,amatrix@couldbeassignedtobetheupper-right2x3blockofBwiththecommand:>Q:=submatrix(B,[1,2],2..4);3-50Q:=|6-21TodefinePtoconsistofcolumns1,3,and4ofrows1and2ofB,use:>P:=submatrix(B,[1,2],[1,3,4]);1-50P=9-21—StudyXY

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ELEMENTARYROWOPERATIONSONMATRICES5Notethattherangenotationcouldnotbeusedtoselectthenonconsecutivecolumns1,3,and4.(ThedefinitionofPcoulduse[1,$(3..4)|asthethirdargumenttothesubmatrixcommand.)Toforma2x4matrixfromtheupper-leftandlower-right2x2blocksofB,constructthetwoblocksseparatelyandthenplacethemside-by-sidewiththecommands:>C1:=submatrix(B,1..2,1..2):>C2:=submatrix(B,2..3,3..4):>C:=augment(C1,C2);[13-21C=9641Theaugmentcommandcreatesacompositematrixbyplacingtheargumentsside-by-sideprovidedeachargumenthasthesamenumberofrows.TostackC1ontopofC2,use:>C3:=stackmatrix(C1,C2);1396C3:=2141.13Exercisesfor§4LigConsiderthematrix08—61342-8203470085~613—7A=23437.-1-10641344433-73008T1437Usematrixsurgerytocreateeachofthe064followingmatrices.5ElementaryRowOperationsonMatricesGaussandGauss-Jordaneliminationcanbecompletedinonestepwiththegausselimandgaussjordancommands,respectively.However,toobtainathoroughunderstandingofthesealgorithmsitisnecessarytopracticedecidingontheelementaryrowoperationtobeperformedateachstep.ThethreeMaplecommandsthatimplementthethreeelementaryrowoperationsareswaprow,mulrow,andaddrow.Toillustrate,considerthetransformationofAtoreducedechelonformwhere0213A=12316].3222—StudyXY

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6MAPLETECHNOLOGYRESOURCEMANUALTheoriginalmatrixis:>A:=matrix(3,4,[0,2,1,3,2,3,1,6,3,2,2,2]1):First,tohaveanon-zeropivotinthefirstrow,interchangerows1and2ofA:>Al:=swaprow(A,1,2);2316Al:=[02133222Thenextstepistocreatealeading1inthefirstrow,multiplyrow1ofAlby1/2:>A2:=mulrow(Al,1,1/A1[1,1]);3112=A2=22’“102133222Toeliminatethe3inthe(1,3)position,add-3timesrow1torow3(ofA2):>A3:=addrow(A2,1,3,-A2(3,1]);311333A43=10213—5Notethatineachsteptheresultoftherowoperationisassignedtoanewname.Whilethisisnotrequired,itisrecommendedsothatthestateateachstepisavailableintheeventitisnecessarytoinvestigateadifferentsequenceofstepsortocorrectanerror.MapleCommandswaprow(A,i,j)|interchangerows4andjofmatrixAmulrow(A,i,r)multiplyrow7ofmatrixAby7addrow(A,j,i,r)|replacerowiofmatrixAwiththesumofrowiand7timesrowjExercisesfor§53.Useelementaryrowoperationstoputthe1.CompletethereductionofmatrixAtofollowingmatrixintoreducedroweche-lonform.reducedrowechelonform.2.CleteExercise44inSection1.2or2.CompleteExercinSection1.2.B=]111132606GraphingCurvesTographtheMapleexpression:>F:=cos(3%x+1)+sin(2¥x-3);F:=cos(3z+1)+sin(2z—3)for0<z<10,use—Study

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GRAPHINGCURVES7>plot(F,x=0..10);187AAYB/|/|i|os|iJ|\SETNPBISBNPUSTof17iiERAN/esq)\VARRVi\/NERYREIi.SaliRR\Toaddaplotofy=sinztothisplot,modifythecommandto:>plot([F,sin(x)J,x=0..10);neAAIAi[[4p[fhAd[A][ERE0.51Por[ABE[ERJEIiTEREgli\lalPRa|e|AEANARBYBRAN4SeRARVARVARIBVAHE:vodfr)|RNLJSY,V4Analternatemethodforcombiningplotsistousethedisplaycommandfromtheplotspackage.Forexample,toaddtheplotsofy=sinandy=coszontheinterval[0,27]tothefirstgraph(onthelargerinterval[0,10])use:>pi:=plot(F,x=0..10):>p2:=plot([cos(x),sin(x)],x=0..2%Pi,color=[GREEN,BLUE]):>display([p1,p2]J);1sAJAAiIfAARSTNJA[\ZNETONRERSPBNPBSla|10NN]\7VoVANS0\Jos;|AdNJ|\/LLYXY/dyorvodDRYy—Study

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8MAPLETECHNOLOGYRESOURCEMANUALNotethattheoutputfromthedefinitionsofp1andp2hasbeensupressed.Toviewoneoftheseplots,sayp2,useeitherdisplay(p2);orp2;.Plotscanbeenhancedwiththeinclusionofatitle,alegend,axislabels,ordifferentlinecolororlinestyle,etc..Theonlinehelptopic?plot,optionsprovidesacompletelistingofallavailableoptions.Toidentifythecoordinatesofapointintheplotwindow,forexample,thefirstintersectionofthegraphsofy=sinzandy=coszintheplotp2,positionthecursoronthedesiredpointandclickthe(left)mousebutton.Thecoordinateofthispointwillappearintheleft-mostboxonthecontextbar.Thesecoordinatescanbecopiedtothedesireddestination.Exercisesfor§6a)Createasingleplotwiththegraphsoff1.Considerthetwofunctionsandg..b)Notethatifdiscont=trueisomittedfromf(z)=cos2z+sinztheplotcommandusedtocreatetheplotandina)thereappeartobesevenplaceswherethetwographsintersect.Explaing(z)=0.0l1tanzNwhytherecanbeonlyfivepointswhereontheinterval0<z<9.Theobjectivethetwographsintersect.Whatcausesofthisproblemistoestimatethepointsthetwo“extraneous”points?wherethegraphsofthesefunctionsin-tersect.c)Estimatethecoordinatesofeachofthefiveintersectionpoints.7MatrixOperationsLetAandBbemxnmatricesandletsbeascalar.TheMaplecommandformatrixadditionisevalm(A+B)andthecommandforscalarmultiplicationisevalm(s*A).ToaddthescalarstoeachelementofamatrixA,usethecommandevalm(A+s).IfAismxnandBisnxp,thematrixproductABisobtainedwiththecommandevalm(A&*B).Asseenpreviously,ifbisann-dimensionalcolumnvector,thecommandlinsolve(A,b)returnsthesetofallsolutionstoAx=b.Toprepareatableoffunctionvaluesfor2z°+3z+1)coszsoy=EETems+sinzforz=1.0,1.2,1.4,...,29.8,30.0,usetheMaplecommands:>f=x=>(x"2+3*x+1)*cos(x)/(2+sin(x));2,fiz(z+3z+1)cos(z)2+sin(z)>X:=[1.41/56$i=10..145]:>Y:=map(f,X):>hdr:=[[x,fx1,[--==—,--—-—-11:>pts:=zip((x,y)->[x,y]1,X,¥Y):>stackmatrix(hdr,pts):Theoutputofthisstackmatrixcommandisa148x2matrix.Intheinterestofconservingspace,thisoutputhasbeensuppressed.The146datapointscanbeplottedwiththecommand:—StudyXY

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TRANSPOSE,NORM,ANDINVERSE9>plot(ptsJ;400A200noooiiFan[HYio{ooofMus|obasia°=JnTTVoREAOBVou—200VdVolVl|iY)Vol—a00byExercisesfor§73.Createatableofvaluesofthefunction1.UseMapletoworkExercises31-41info)=tan(1+2?)Section1.5.x)=(1+z3)itz2.UseMaple’slinsolvecommandtosolveatthepointsz=2.0,2.1,...,5.0.thesysteminExercise27inSection1.1.8Transpose,Norm,andInverseTodefineB=A",typeB:=evalm(transpose(A));.Maple’snormcommandcomputesthenormofamatrixorvector.ToobtaintheEuclideannormofavector,besuretoinclude2asthesecondargument,e.g.,norm(x,2).Tofindtheinverseofasquarematrix4,usetheMaplecommandinverse(A).TheinverseiscalculatedusingGaussian-JordaneliminationwhenAislargerthan4x4;forsmallermatrices,determinantsandCramer’sruleareused.IfanyentryofAisafloating-pointnumber,thenMaplecomputesA!asafloating-pointmatrix.Inallothercases,e.g.,allentriesofAareintegersorrationalnumbers,MaplewillcomputeanexactrepresentationforA1.IfAissingular,ornumericallysingular,Maplewillreportthiswithanerrormessage.—StudyXY

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10MAPLETECHNOLOGYRESOURCEMANUALExercisesfor§8a)ExecutetheMaplecommandsinverse(A);1.UseMapletofindtheinverseofthema-andinverse(evalf(evalm(A))):.NotethatA©.inbothcasesMaplereportsthatthema-trixinExercise20ofSection1.9.ewtrixissingular.2.UseMapletofindtheinverseofthema-.trixinExercise27ofSection1.9.b)Repeata)forthematrix5.3.ToillustrateMaple’shandlingofsingu-©)Repeata)forthematrixCfori=—1,y.—2,...,—20.(Hint:UsetheMaplecom-larmatricesandthedifferencesbetweenmandinverse(eval(evalm(C),i=-1)):.)exactandfloating-pointarithmetic,con-Bensiderthematricesd)Repeat«c¢)whentheentriesof3694Carefloating-pointnumbers.A=19,B=12]s(Hint:UsetheMaplecommandinverse(evalf(eval(evalm(C),i=-1)));.)c=12+10°e)Explainthedifferentresultsinpartsc)12:andd).f)(Optional.)ComparetheseresultswiththoseobtainedwithMATLAB.9SpecialMatricesTodefinethezeromatrix,usetheMaplecommandmatrix(m,n,0)wheremandnarethenumberofrowsandcolumns,respectively,inA.Likewise,matrix(2,3,1)createsa2x3matrixinwhicheveryentryis1.Moregenerally,B:=matrix(m,n,f);createsanmxnmatrixBwithentriesbij=f(i,7).Toillustrate,the3x3matrixinwhichtheentriesalternatebetween1and0,startingwith1,canbecreatedwiththecommandB:=matrix(3,3,(i,j)->i+j+1mod2);.The5x5identitymatrixisobtainedwithmatrix(5,5,(i,j)—>ifi=jthen1else0endif).ArandommxnmatrixcanbecreatedwiththeMaplecommandrandmatrix(m,n).Bydefault,theentriesinthismatrixrangefrom-99to99,thatisatwo-digitinteger.Tolimitthevaluestointegersbetweenaandb,inclusive,addathirdargument:randmatrix(m,n,entries=rand(a..b)).Randommatricesarefrequentlyusedtotestaconjectureorsimulateaseriesofrandomevents,suchastrafficflowinastreetnetwork.Exercisesfor§9b)Usethegaussjordcommandtosolveeach1.Createa4x5matrixconsistingentirelyofthe10systems.ofsevens.c)Usingpencilandpaper(orlinsolve),5writeeachsolutioninstandardform.:Whatcanyouconcludeifm<n?(Hint:a)UsetherandmatrixcommandtogenerateSeeTheorem4,Section1.3.)10differenthomogeneousmxnsystemswithintegerentriesbetween-20and20.Makesureatleast6ofthematricesm<n.—StudyXY

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WORKINGWITHDATAINMAPLE1110WorkingwithDatainMapleMaplehasanumberoftoolsforworkingwithlistsofdata.The(minimal)interpolatingpolynomialforacollectionofdatapointscanbefoundwiththeinterpcommand.Also,thestatspackagecontainsacommand,fit,thatcandeterminethebestleast-squaresfitofdatatoagivenfamilyoffunctions.Toillustratetheuseofthesecommands,considerthetableofdatapoints:z|123456y|41217334552Thelistsofzandyvaluesare:>xi:=[8$1..61];21:=[1,2,3,4,5,6|>yi:=[4,12,17,33,45,521;ul=[4,12,17,33,45,52)Thecorrespondingdatapointsare:>pts:=zip((x,y)->[x,y],x1,yi);pts=[[1,4],[2,12],[3,17],[4,33],[5,45},[6,52]]Forfuturereference,constructaplotofthedatapoints:>pt_plot:=pointplot(pts):Thebestleast-squareslinearfittothedatais:>1s_fit:=fit[leastsquare[[x,y],y=a*x+b,{a,b}11([x1,y1]);7125Is_fity=roThefollowinggraphprovidesavisualconfirmationofthisresult:>1s_plot:=plot(rhs(ls_fit),x=0..7):>display([pt_plot,1ls_plot]);so~~Pp=ABSEPBEaoaThebestleast-squaresquinticfittothese6datapointsisthepolynomialinterpolator:>fit[leastsquarel[x,y],y=a*x"5+b*x~4+c*x"3+d*x"2+exx+f,{a,b,c,d,e,f}1]>C[x1,y1]J;43,79,359,1661,5903¥=1557prtgEToz="30z—93—StudyXY

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12MAPLETECHNOLOGYRESOURCEMANUALbutitissimplertoobtainthisinterpolatingpolynomialdirectlyfromthedatapoints:>interp_poly:=interp(x1,yi,xJ);437935916615903interp_poly=—5—2pd4227132—interp_poly150°Bot3x5z°430z—93Thebestlinearfitandinterpolatingpolynomialareeasilycomparedinagraph:>ip_plot:=plot(interp_poly,x=0..7,color=GREEN):>display([pt_plot,ls_plot,ip_plot]);100‘°esNotethatthedataconsistsofintegersandthecoefficientscomputedforthebestleast-squareslinearfitandinterpolatingpolynomialarerational;nofloating-pointapproximationshavebeenem-ployed.Toseefloating-pointapproximationstothesepolynomials,changeoneormoredatavaluetoafloating-pointnumberoruseevalf(interp_poly);.Exercisesfor§10c)Plotthedatapoints,bestleast-squares1.Considerthefollowingtableofdatafit,andinterpolatoronthesamegraph.2.Repeat1.withthetableofdataz|12345y[33526790108z|{-3-2-10123a)Findthebestleast-squaresfittothey{337.810512.2109823.1data.b)Findthepolynomialinterpolatorforthedata.11ProceduresTheMapleworksheetprovidesaconvenientenvironmentfordevelopingandarchivingsequencesofMaplecommandsthatwillbeexecutedmanytimes.ItisalsopossibletoreadMaplecommandsfromatextfile;see7readforacompleteexplanation.Tofurtherfacilitatetheimplementationofnewalgorithms,blocksofMaplecommandscanbegroupedtogethertodefineaprocedure.(AninterestingfactaboutMapleisthatthevastmajorityoftheMaplecommandsarewritteninMaple—asprocedures.)Forexample,supposeitisnecessarytocomparebestleast-squarespolynomialfitstoagivensetofdatapointsforpolynomialsofdifferentdegrees.Thefollowingprocedure,namedIsplot,[+s

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PROCEDURES13threearguments—thelistof«values,thelistofyvalues,andthedegreeofthepolynomialtobeusedintheleast-squaresfit—andreturnsaplotofthedatapointsandthebestleast-squrarepolynomialoftherequesteddegree:>1splot:=proc(X,Y,n)>locala,ls_plot,p,pts,pt_plot,x,y;>global1ls_fit;>p:=add(alli*xx™i,i=0..n);>ai={coeffs(p,x)};>1s_fit:=fit[leastsquarel[[x,y],y=p,a11([X,Y]);>pts:=zip((x,y)->[x,y],X,Y);>pt_plot:=pointplot(pts,color=RED);>1s_plot:=plot(rhs(Is_fit),x=min(op(X))-1..max(op(X))+1,>color=GREEN);>returndisplay([pt_plot,1ls_plot]);>endproc:Thisprocedurecloselyfollowstheapproachpresentedintheprevioussection.ThefirstlinedefinesIsplottobeaMapleprocedureandspecifiesthearguments.Line2,beginninglocal,definesthenamesthataretoremainlocaltothisprocedure;thenextlinedefinesIsfittobeaglobalvariable.Localvariablesdonotassumeanyvalueatthestartoftheprocedureandtheirfinalvaluesarenotaccessibleoncetheprocedureterminates.Conversely,globalvariablesretaintheirvalueaftertheprocedureterminatesandcanbeaccessedfromelsewhereintheMaplesession.(Foradditionalinformationonlocalandglobalvariablessee?procedure;see?parameterforparameterpassinginMapleprocedures.)Thenexttwolinesdefineptobethegeneralpolynomialofdegreenandatobethesetofcoefficientsinp.ThenextfourlinesimplementthealgorithmdiscussedinSection10.Notethatthepolynomialisplottedoveranintervalthatextendsoneunitoneithersideofthesmallestandlargestzvaluewithoutassumingthatthesevaluesaresorted.ThefinalcommandinstructsMapletoreturnthecombinedplotastheresultoftheIsplotcommand.Now,thebestleast-squarescubicfittothedatainSection10canbegraphedviathesinglecommand:>1splot(x1,yi,3);so=/“0,20,se0.5g=ER5s>Becausethebestleast-squaresfitisaglobalvariable,itcanbeaccessedasifithadbeenreturnedbytheprocedure:>1s_fit;_%3355MB,554VCETERTm108AnotherwaytodefineaprocedureistouseMaple’s“arrow”notation(see?->).Thisformispar-ticularlysuitedtotheimplementationofmathematicalfunctionsandcommandsthatareexpressed—StudyXY

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14MAPLETECHNOLOGYRESOURCEMANUALwithouttheneedforlocalvariablesorintermediateresults.Forexample,thescalartripleproductofthevectorsu,v,andw,u-(vxw),canberepresentedas:>sc_triple_prod:=(u,v,w)->dotprod(u,crossprod(v,w));sc_triple_prod:=(u,v,w)—dotprod(u,crossprod(v,w))Considerthevectors:>a:=vector([1,2,3]):>b:=vector([2,3,1]):>¢:=vector([1,2,1]):Thescalartripleproductofa,b,andcis>sc_triple_prod(a,b,cJ;2Exercisesfor§112.UsetheprocedureIsplotwithavalueofn-3)i1.WriteaMapleprocedurethatcalculates0thatthebestleast-squarespolynomial.approximationofdegreenorlesslooksthevectortripleproductux(vxw).5Testfunctionusingthevectorsreasonableforthedatapoints(z;,y;)=estyourfunctionusing(ini),i=1,2,...,20.(Hint:Thereis19010needtoconsidern>5.)a=|1|,b=|-2],e=]|1],214—StudyXY

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