Introduction-to-Linear-Algebra(第四版答案)-修改版

INTRODUCTIONTOLINEARALGEBRAFourthEditionMANUALFORINSTRUCTORSGilbertStrangMassachusettsInstituteofTechnologymath.mit.edu/linearalgebraweb.mit.edu/18.06videolectures:ocw.mit.edumath.mit.edu/[e-mail]-CambridgePressBox812060Wellesley,Massachusetts024822SolutionstoExercisesProblemSet1.1,page81Thecombinationsgive(a)alineinR3(b)aplaneinR3(c)allofR3.2vCwD.2;3/andvwD.6;1/willbethediagonalsoftheparallelogramwithvandwastwosidesgoingoutfrom.0;0/.3ThisproblemgivesthediagonalsvCwandvwoftheparallelogramandasksforthesides:TheoppositeofProblem2.InthisexamplevD.3;3/andwD.2;2/.43vCwD.7;5/andcvCdwD.2cCd;cC2d/.5uCvD.2;3;1/anduCvCwD.0;0;0/and2uC2vCwD.addfirstanswers/D.2;3;1/.Thevectorsu;v;wareinthesameplanebecauseacombinationgives.0;0;0/.Statedanotherway:uDvwisintheplaneofvandw.6ThecomponentsofeverycvCdwaddtozero.cD3anddD9give.3;3;6/.7Theninecombinationsc.2;1/Cd.0;1/withcD0;1;2anddD.0;1;2/willlieonalattice.Ifwetookallwholenumberscandd,thelatticewouldlieoverthewholeplane.8Theotherdiagonalisvw(orelsewv).Addingdiagonalsgives2v(or2w).9Thefourthcornercanbe.4;4/or.4;0/or.2;2/.Threepossibleparallelograms!10ijD.1;1;0/isinthebase(x-yplane).iCjCkD.1;1;1/istheoppositecornerfrom.0;0;0/.Pointsinthecubehave0x1,0y1,0z1.11Fourmorecorners.1;1;0/;.1;0;1/;.0;1;1/;.1;1;1/.Thecenterpointis.12;12;12/.Centersoffacesare.12;12;0/;.12;12;1/and.0;12;12/;.1;12;12/and.12;0;12/;.12;1;12/.12Afour-dimensionalcubehas24D16cornersand2
4D8three-dimensionalfacesand24two-dimensionalfacesand32edgesinWorkedExample2.4A.13SumDzerovector.SumD2:00vectorD8:00vector.2:00is30ıfromhorizontalD.cos6;sin6/D.p3=2;1=2/.14Movingtheoriginto6:00addsjD.0;1/toeveryvector.Sothesumoftwelvevectorschangesfrom0to12jD.0;12/.15Thepoint34vC14wisthree-fourthsofthewaytovstartingfromw.Thevector14vC14wishalfwaytouD12vC12w.ThevectorvCwis2u(thefarcorneroftheparallelogram).16AllcombinationswithcCdD1areonthelinethatpassesthroughvandw.ThepointVDvC2wisonthatlinebutitisbeyondw.17AllvectorscvCcwareonthelinepassingthrough.0;0/anduD12vC12w.ThatlinecontinuesoutbeyondvCwandbackbeyond.0;0/.Withc0,halfofthislineisremoved,leavingaraythatstartsat.0;0/.18ThecombinationscvCdwwith0c1and0d1filltheparallelogramwithsidesvandw.Forexample,ifvD.1;0/andwD.0;1/thencvCdwfillstheunitsquare.19Withc0andd0wegettheinfinite“cone”or“wedge”betweenvandw.Forexample,ifvD.1;0/andwD.0;1/,thentheconeisthewholequadrantx0,y0.Question:WhatifwDv?Theconeopenstoahalf-space.SolutionstoExercises320(a)13uC13vC13wisthecenterofthetrianglebetweenu;vandw;12uC12wliesbetweenuandw(b)Tofillthetrianglekeepc0,d0,e0,andcCdCeD1.21Thesumis.vu/C.wv/C.uw/Dzerovector.Thosethreesidesofatriangleareinthesameplane!22Thevector12.uCvCw/isoutsidethepyramidbecausecCdCeD12C12C121.23Allvectorsarecombinationsofu;v;wasdrawn(notinthesameplane).StartbyseeingthatcuCdvfillsaplane,thenaddingewfillsallofR3.24Thecombinationsofuandvfilloneplane.Thecombinationsofvandwfillanotherplane.Thoseplanesmeetinaline:onlythevectorscvareinbothplanes.25(a)Foraline,chooseuDvDwDanynonzerovector(b)Foraplane,chooseuandvindifferentdirections.AcombinationlikewDuCvisinthesameplane.26Twoequationscomefromthetwocomponents:cC3dD14and2cCdD8.ThesolutioniscD2anddD4.Then2.1;2/C4.3;1/D.14;8/.27ThecombinationsofiD.1;0;0/andiCjD.1;1;0/fillthexyplaneinxyzspace.28Thereare6unknownnumbersv1;v2;v3;w1;w2;w3.ThesixequationscomefromthecomponentsofvCwD.4;5;6/andvwD.2;5;8/.Addtofind2vD.6;10;14/sovD.3;5;7/andwD.1;0;1/.29TwocombinationsoutofinfinitelymanythatproducebD.0;1/are2uCvand12w12v.No,threevectorsu;v;winthex-yplanecouldfailtoproducebifallthreelieonalinethatdoesnotcontainb.Yes,ifonecombinationproducesbthentwo(andinfinitelymany)combinationswillproduceb.ThisistrueevenifuD0;thecombinationscanhavedifferentcu.30Thecombinationsofvandwfilltheplaneunlessvandwlieonthesamelinethrough.0;0/.Fourvectorswhosecombinationsfill4-dimensionalspace:oneexampleisthe“standardbasis”.1;0;0;0/;.0;1;0;0/;.0;0;1;0/,and.0;0;0;1/.31TheequationscuCdvCewDbare2cdD1cC2deD0dC2eD0SodD2ethencD3ethen4eD1cD3=4dD2=4eD1=4ProblemSet1.2,page191u
vD1:8C3:2D1:4,u
wD4:8C4:8D0,v
wD24C24D48Dw
v.2kukD1andkvkD5andkwkD10.Then1:4.1/.5/and48.5/.10/,confirmingtheSchwarzinequality.3Unitvectorsv=kvkD.35;45/D.:6;:8/andw=kwkD.45;35/D.:8;:6/.Thecosineofisvkvk
wkwkD2425.Thevectorsw;u;wmake0ı;90ı;180ıangleswithw.4(a)v
.v/D1(b).vCw/
.vw/Dv
vCw
vv
ww
wD1C././1D0soD90ı(noticev
wDw
v)(c).v2w/
.vC2w/Dv
v4w
wD14D3.4SolutionstoExercises5u1Dv=kvkD.3;1/=p10andu2Dw=kwkD.2;1;2/=3.U1D.1;3/=p10isperpendiculartou1(andsois.1;3/=p10).U2couldbe.1;2;0/=p5:Thereisawholeplaneofvectorsperpendiculartou2,andawholecircleofunitvectorsinthatplane.6AllvectorswD.c;2c/areperpendiculartov.Allvectors.x;y;z/withxCyCzD0lieonaplane.Allvectorsperpendicularto.1;1;1/and.1;2;3/lieonaline.7(a)cosDv
w=kvkkwkD1=.2/.1/soD60ıor=3radians(b)cosD0soD90ıor=2radians(c)cosD2=.2/.2/D1=2soD60ıor=3(d)cosD1=p2soD135ıor3=4.8(a)False:vandwareanyvectorsintheplaneperpendiculartou(b)True:u
.vC2w/Du
vC2u
wD0(c)True,kuvk2D