mit线性代数problem set 6

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1、18.06ProblemSet6SolutionsTotal:100pointsSection4.3.Problem4:WritedownE=�Ax−b�2asasumoffoursquares–thelastoneis(C+4D−20)2.Findthederivativeequations∂E/∂C=0and∂E/∂D=0.Divideby2toobtainthenormalequationsATAx�=ATb.Solution(4points)Observe⎛⎞⎛⎞100��⎜⎜11⎟⎟⎜⎜8⎟⎟

2、CA=,b=,anddefinex=.⎝13⎠⎝8⎠D1420Then⎛⎞C⎜⎜C+D−8⎟⎟Ax−b=,⎝C+3D−8⎠C+4D−20and22222�Ax−b�=C+(C+D−8)+(C+3D−8)+(C+4D−20).Thepartialderivativesare∂E/∂C=2C+2(C+D−8)+2(C+3D−8)+2(C+4D−20)=8C+16D−72,∂E/∂D=2(C+D−8)+6(C+3D−8)+8(C+4D−20)=16C+52D−224.Ontheotherhand,����T48

3、T36AA=,Ab=.826112Thus,ATAx=ATbyieldstheequations4C+8D=36,8C+26D=112.Multiplyingby2andlookingback,weseethatthesearepreciselytheequations∂E/∂C=0and∂E/∂D=0.Section4.3.Problem7:Findtheclosestlineb=Dt,throughtheorigin,tothesamefourpoints.AnexactfitwouldsolveD·

4、0=0,D·1=8,D·3=8,D·4=20.1Findthe4by1matrixAandsolveATAx�=ATb.Redrawfigure4.9ashowingthebestlineb=Dtandthee’s.Solution(4points)Observe⎛⎞⎛⎞00⎜⎜1⎟⎟⎜⎜8⎟⎟TTA=,b=,AA=(26),Ab=(112).⎝3⎠⎝8⎠420Thus,solvingATAx=ATb,wearriveatD=56/13.Hereisthediagramanalogoustofigure4.

5、9a.Section4.3.Problem9:Formtheclosestparabolab=C+Dt+Et2tothesamefourpoints,andwritedowntheunsolvableequationsAx=binthreeunknowns2x=(C,D,E).SetupthethreenormalequationsATAx�=ATb(solutionnotrequired).Infigure4.9ayouarenowfittingaparabolato4points–whatishappe

6、ninginFigure4.9b?Solution(4points)Note⎛⎞⎛⎞1000⎛⎞⎜111⎟⎜8⎟CA=⎜⎟,b=⎜⎟,x=⎝D⎠.⎝139⎠⎝8⎠E141620ThenmultiplyingoutAx=byieldstheequationsC=0,C+D+E=8,C+3D+9E=8,C+4D+16E=20.Takethesumofthefourthequationandtwicethesecondequationandsubtractthesumofthefirstequationandt

7、wotimesthethirdequation.Onegets0=20.Hence,theseequationsarenotsimultaneouslysolvable.Computing,weget⎛⎞⎛⎞482636TTAA=⎝82692⎠,Ab=⎝112⎠.2692338400Thus,solvingthisproblemisthesameassolvingthesystem⎛⎞⎛⎞⎛⎞4826C36⎝82692⎠⎝D⎠=⎝112⎠.2692338E400Theanalogueofdiagram4

8、.9(b)inthiscasewouldshowthreevectorsa1=(1,1,1,1),a=(0,1,3,4),a=(0,1,9,16)spanningathreedimensionalvectorsubspaceofR4.23Itwouldalsoshowthevectorb=(0,8,8,20),andtheprojectionp=Ca1+Da2+Ea3ofbintothethreedimensionalsubspace.Se