MIT英文版线性代数试卷(1)

1�18.06Quiz1 March1,2010ProfessorStrangYourPRINTEDnameis:1.Yourrecitationnumberorinstructoris2.3.4.1.ForwardeliminationchangesAx=btoarowreducedRx=d:thecompletesolutionis⎤⎡⎤⎡⎤⎡425x=⎢⎢⎢⎣⎥⎥⎥⎦+c1⎢⎢⎢⎣⎥⎥⎥⎦+c2⎢⎢⎢⎣⎥⎥⎥⎦010001(a) (14points)Whatisthe3by3reducedrowechelonmatrixRandwhatisd?Solution:First,sinceRisinreducedrowechelonform,wemusthave�Td=400TheothertwovectorsprovidespecialsolutionsforR,showingthatRhasrank1:again,sinceitisinreducedrowechelonform,thebottomtworowsmustbeall0,and⎡⎤R=��Tthetoprowis1−2−5,i.e.⎢⎢⎢⎣10−20−50⎥⎥⎥⎦.000(b) (10points)Iftheprocessofeliminationsubtracted3timesrow1fromrow2andthen5timesrow1fromrow3,whatmatrixconnectsRanddtotheoriginalAandb?UsethismatrixtofindAandb.Solution:ThematrixconnectingRanddtotheoriginalAandbis⎤⎡⎤⎡=⎤ ⎥⎥⎥⎦⎡ ⎢⎢⎢⎣100−310−501100100⎢⎢⎢⎣⎥⎥⎥⎦·⎢⎢⎢⎣⎥⎥⎥⎦E=E31E21=010−310−501001Thatis,R=EAandEb=d.Thus,A=E−1Randb=E−1d,giving⎤⎡⎤⎡1 001−2−5A=3 10000=·⎡⎤⎢⎢⎢⎣1−23−6−5−15⎥⎥⎥⎦5−10−25⎡⎤⎢⎢⎢⎣412⎥⎥⎥⎦20⎢⎢⎢⎣⎥⎥⎥⎦⎢⎢⎢⎣⎥⎥⎥⎦501000⎤⎡⎤⎡1004⎢⎢⎢⎣⎥⎥⎥⎦·⎢⎢⎢⎣⎥⎥⎥⎦b=3100=50102.SupposeAisthematrix⎤⎡0122⎢⎢⎢⎣⎥⎥⎥⎦A=0387.0042(a) (16points)FindallspecialsolutionstoAx=0anddescribeinwordsthewholenullspaceofA.Solution:First,byrowreduction⎤⎡⎤⎡⎤⎡⎤⎡01220122010101011⎢⎢⎢⎣⎥⎥⎥⎦→⎢⎢⎢⎣⎥⎥⎥⎦→⎢⎢⎢⎣⎥⎥⎥⎦→⎢⎢⎢⎣⎥⎥⎥⎦03870021002100120042004200000000sothespecialsolutionsare⎤⎥⎥⎥⎥⎥⎥⎦0−1−121⎡⎥⎥⎥⎥⎥⎥⎦,s2=⎢⎢⎢⎢⎢⎢⎣⎤1000⎡s1=⎢⎢⎢⎢⎢⎢⎣Thus,N(A)isaplaneinR4givenbyalllinearcombinationsofthespecialsolutions.(b) (10points)DescribethecolumnspaceofthisparticularmatrixA.“Allcombinationsofthefourcolumns”isnotasufficientanswer.Solution:C(A)isaplaneinR3givenbyallcombinationsofthepivotcolumns,namely⎤⎥⎥⎥⎦⎡⎢⎢⎢⎣⎤⎥⎥⎥⎦⎡⎢⎢⎢⎣12c13+c2804(c)(10points)WhatisthereducedrowechelonformR∗=rref(B)whenBisthe6by8blockmatrix⎤⎡AAB=⎦usingthesameA?AASolution:NotethatBimmediatelyreducesto⎣⎤⎡AAB=⎦⎣00WereducedAabove:therowreducedechelonformofofBisthus⎤⎥⎥⎥⎦⎡⎢⎢⎢⎣01011B=rref(A)rref(A)⎦,rref(A)=0012⎤000000⎡⎣3.(16points)Circlethewordsthatcorrectlycompletethefollowingsentence:(a)Supposea3by5matrixAhasrankr=3.ThentheequationAx=b(always/sometimesbutnotalways)has(auniquesolution/manysolutions/nosolution).Solution:theequationAx=balwayshasmanysolutions.(b)WhatisthecolumnspaceofA?DescribethenullspaceofA.Solution:Thecolumnspaceisa3-dimensionalspaceinsidea3-dimensionalspace,i.e.itcontainsallthevectors,andthenullspacehasdimension5−3=20insideR5.4.SupposethatAisthematrix⎤⎡21A=65.⎢⎢⎢⎣⎥⎥⎥⎦24(a) (10points)ExplaininwordshowknowingallsolutionstoAx=bdecidesifagivenvectorbisinthecolumnspaceofA.Solution:ThecolumnspaceofAcontainsalllinearcombinationsofthecolumnsofA,whicharepreciselyvectorsoftheformAxforanarbitraryvectorx.Thus,Ax=bhasasolutionifandonlyifbisinthecolumnspaceofA.⎤⎡8⎢⎢⎢⎣⎥⎥⎥⎦inthecolumnspaceofA?(b)(14points)Isthevectorb=2814Solution:Yes.ReducingthematrixcombiningAandbgives⎤⎡⎤⎡⎤⎡⎢⎢⎢⎣2 18⎥⎥⎥⎦→⎢⎢⎢⎣2 18⎥⎥⎥⎦→⎢⎢⎢⎣2 18⎥⎥⎥⎦6 5280 240 242 414036000⎤⎡⎣3⎦isasolutiontoAx=b,andbisinthecolumnspaceofA.Thus,x=2MITOpenCourseWare: