introduction to linear algebra solution manual

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1、INTRODUCTIONTOLINEARALGEBRAFourthEditionMANUALFORINSTRUCTORSGilbertStrangMassachusettsInstituteofTechnologymath.mit.edu/linearalgebraweb.mit.edu/18.06videolectures:ocw.mit.edumath.mit.edu/gswww.wellesleycambridge.comemail:gs@math.mit.eduWellesley-CambridgePressBox812060Wellesley

2、,Massachusetts024822SolutionstoExercisesProblemSet1.1,page81Thecombinationsgive(a)alineinR3(b)aplaneinR3(c)allofR3.2vCwD.2;3/andvwD.6;1/willbethediagonalsoftheparallelogramwithvandwastwosidesgoingoutfrom.0;0/.3ThisproblemgivesthediagonalsvCwandvwoftheparallelogramandasksforthe

3、sides: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.6Thecomponentso

4、feverycvCdwaddtozero.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

5、.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.1;1;1/.222Centersoffacesare.1;1;0/;.1;1;1/and.0;1;1/;.1;1;1/and

6、.1;0;1/;.1;1;1/.22222222222212Afour-dimensionalcubehas24D16cornersand2
4D8three-dimensionalfacesand24two-dimensionalfacesand32edgesinWorkedExample2.4A.13SumDzerovector.SumD2:00vectorD8:00vector.2:00is30ıfromhorizontalpD.cos;sin/D.3=2;1=2/.6614Movingtheoriginto6:00addsjD.0;1/to

7、everyvector.Sothesumoftwelvevectorschangesfrom0to12jD.0;12/.3115ThepointvCwisthree-fourthsofthewaytovstartingfromw.Thevector441111vCwishalfwaytouDvCw.ThevectorvCwis2u(thefarcornerofthe4422parallelogram).16AllcombinationswithcCdD1areonthelinethatpassesthroughvandw.ThepointVDvC2wi

8、sonthatlinebutitisbeyondw.17Allvectorscv