经济学中的数学Mathematics for economists课后答案answers2

ANSWERSPAMPHLET51b)x(s,t)50@0321A10@30211At10@22221As,x12y13z512.10.37a)Rewritingthesymmetricequations,y5y02(b6a)x01(b6a)xandz5z02(c6a)x01(c6a)x.Then0@x(t)y(t)z(t)1A50@x0y0z01A10@1b6ac6a1Ator0@x(t)y(t)z(t)1A50@x0y0z01A10@abc1At.b)Thetwoplanesaredescribedbyanytwodistinctequalitiesinsystem(20).Forexample,x2x0a5y2y0bandy2y0b5z2z0c.Inotherwords,2bx1ay52bx01ay0andcy2bz5cy02bz0.c)i)x122215x22345x3215,ii)x12145x22255x3236.d)i)4x11x2511and5x224x3511.ii)5x124x2523and6x225x3523.10.38a)Normals(1,2,23)and(1,3,22)donotlineup,soplanesintersect.b)Normals(1,2,23)and(22,24,6)dolineup,soplanesdonotintersect.10.39a)(x21,y22,z23)?(21,1,0)50,sox2y521.b)Thelinerunsthroughthepoints(4,2,6)and(1,3,11),sotheplanemustbeorthogonaltothedifferencevector(23,1,5).Thus(x21,y21,z11)?(23,1,5)50,or23x1y15z527.c)Thegeneralequationfortheplaneisax1by1gz5d.Theequationstobesatisfiedarea5d6a,b5d6b,andc5d6g.Asolutionisa516a,b516b.g516candd51,so(16a)x1(16b)y1(16c)z51.10.40Plugx531t,y5127t,andz5323tintotheequationx1y1z51oftheplaneandsolvefort:(31t)1(127t)1(323t)51=)t5263.Thepointis¡1163,21163,1¢.10.410@1121...4121...31A=)0@1023...5012...211A=)x5513zy52122z.Takingz5t,writethelineas0@xyz1A50@52101A1t0@32211A.52MATHEMATICSFORECONOMISTS10.42IS:[12c1(12t1)2a0]Y1(a1c2)r5c02c1t01Ip1GLM:mY2hr5Ms2Mp.Iprises=)ISmovesup=)Ypandrpincrease.Msrises=)LMmovesup=)Ypdecreasesandrpincreases.mrises=)LMbecomessteeper=)Ypdecreasesandrpincreases.hrises=)LMflatter=)Ypincreasesandrpdecreases.a0rises=)ISflatter(withsamer-intercept)=)rpandYprise.c0rises=)ISmovesup=)Ypandrpincrease.t1ort2rises=)ISsteeper(withsamer-intercept)=)rpandYpdecrease.Chapter1111.1Supposev15r2v2.Then1v12r2v25r2v22r2v250.Supposec1v11c2v250.SupposethatciÞ0.Thencivi52cjvj,sovi5(cj6ci)vj.11.2a)Condition(3)givestheequationsystem2c11c250c112c250.Theonlysolutionisc15c250,sothesevectorsareindependent.b)Condition(3)givestheequationsystem2c11c25024c122c250.Onesolutionisc1522,c251,sothesevectorsaredependent.c)Condition(3)givestheequationsystemc150c11c250c21c350.Clearlyc15c250istheuniquesolution.ANSWERSPAMPHLET53d)Condition(3)givestheequationsystemc11c350c11c250c21c350.Theonlysolutionisc15c25c350,sothesevectorsareindependent.11.3a)Thecoeffi[e-mail]3,sothehomogeneousequationsystemhasonlyonesolution,c15c25c350.Thusthesevectorsareindependent.b)Thecoeffi[e-mail]2,sothehomogeneousequationsystemhasaninfinitenumberofsolutionsasidefromc15c25c350.Thusthesevectorsaredependent.11.4Ifc1v15v2,thenc1v11(21)v250,andcondition(4)failstohold.Ifcondition(4)failstohold,thenc1v11c2v250hasanonzerosolutioninwhich,say,c2Þ0.Thenv25(c16c2)v1,andv2isamultipleofv1.11.5a)Thenegationof“c1v11c2v21c3v350impliesc15c25c350”is“Thereissomenonzerochoiceofc1,c2,andc3thatc1v11c2v21c3v350.”b)Supposethatcondition(5)failsandthatc1Þ0.Thenv152(c26c1)v12(c36c1)v3.11.6Supposehv1,...,vnjisacollectionofvectorssuchthatv150.Then1v110v21???10vn50andthevectorsarelinearlydependent.11.7A?0B@c1...ck1CA5c1v11???1ckvk,sothecolumnsofAarelinearlyindependentifandonlyiftheequationsystemA?c50hasnononzerosolution.54MATHEMATICSFORECONOMISTS11.8TheconditionofTheorem2isnecessaryandsufficientfortheequationsystemofTheorem1tohaveauniquesolution,whichmustbethetrivialsolution.ThusAhasrankn,anditfollowsfromTheorem9.3thatdetAÞ0.11.9a)(2,2)53(1,2)21(1,4).b)[e-mail]150,c251,andc352.11.10Notheydonot.CheckingtheequationsystemAx5b,weseethatAhasrank2;thismeansthattheequationsystemdoesnothaveasolutionforgeneralb.11.11Foranycolumnvectorb,iftheequationsystemAx5bhasasolutionxp,thenxp1v11???1xpnvn5b.Consequently,iftheequationsystemhasasolutionforeveryright-handside,theneveryvectorbcanbewrittenasalinearcombinationofthecolumnvectorsvi.Conversely,iftheequationsystemfailstohaveasolutionforsomeright-handsideb,thenbisnotalinearcombinationofthevi,andthevidonotspanRn.11.12Thevectorsinaarenotindependent.Thevectorsinbareabasis.Thevectorsincarenotindependent.Thevectorsindareabasis.11.13a)detµ1001¶51Þ0,sothesevectorsspanR2andareindependent.b)detµ21001¶521Þ0,sothesevectorsspanR2andareindependent.d)detµ11211¶52Þ0,sothesevectorsspanR2andareindependent.11.14TwovectorscannotspanR3,sothevectorsinaarenotabasis.MorethanthreevectorsinR3cannotbeindependent,sothevectorsinefailtobeabasis.Thematrixofcolumnvectorsinbandchaveranklessthan3,soneitherofthesecollectionsofvectorsisabasis.The333matrixAwhosecolumnvectorsarethevectorsofdhasrank3,sodetAÞ0.ThustheyareabasisaccordingtoTheorem11.8.11.15Supposeaistrue.Thenthen3nmatrixAwhosecolumnsarethevectorsvihasrankn,andthereforeAx5bhasasolutionforeveryright-handsideb.ThusthecolumnvectorsvispanRn.SincetheyarelinearlyindependentandspanR3,theyareabasis.SincetherankofAisn,detAÞ0.ThusaANSWERSPAMPHLET55impliesb,candd.Finallysupposedistrue.SincedetAÞ0,thesolutionx50toAx50isunique.ThusthecolumnsofAarelinearlyindependent;dimpliesa.Chapter1212.1a)xn5n.b)xn516n.c)xn52(21)n21(n21).d)x