MIT18_06SCF11_Ses1.6Transposes, Permutations, Vector Spaces

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1、ColumnspaceandnullspaceInthislecturewecontinuetostudysubspaces,particularlythecolumnspaceandnullspaceofamatrix.ReviewofsubspacesAvectorspaceisacollectionofvectorswhichisclosedunderlinearcombinations.Inotherwords,foranytwovectorsvandwinthespaceandanytworealnumberscandd,thevectorcv+dwisalsointhe

2、vectorspace.Asubspaceisavectorspacecontainedinsideavectorspace.����00AplanePcontaining0andalineLcontaining0arebothsub00spacesofR3.TheunionP∪Lofthosetwosubspacesisgenerallynotasubspace,becausethesumofavectorinPandavectorinLisprobablynotcontainedinP∪L.TheintersectionS∩ToftwosubspacesSandTisasubs

3、pace.Toprovethis,usethefactthatbothSandTareclosedunderlinearcombinationstoshowthattheirintersectionisclosedunderlinearcombinations.ColumnspaceofAThecolumnspaceofamatrixAisthevectorspacemadeupofalllinearcombinationsofthecolumnsofA.SolvingAx=bGivenamatrixA,forwhatvectorsbdoesAx=bhaveasolutionx?⎡

4、⎤112⎢213⎥LetA=⎢⎥.⎣314⎦415ThenAx=bdoesnothaveasolutionforeverychoiceofbbecausesolvingAx=bisequivalenttosolvingfourlinearequationsinthreeunknowns.IfthereisasolutionxtoAx=b,thenbmustbealinearcombinationofthecolumnsofA.Onlythreecolumnscannotfilltheentirefourdimensionalvectorspace–somevectorsbcannot

5、beexpressedaslinearcombinationsofcolumnsofA.Bigquestion:whatb’sallowAx=btobesolved?Ausefulapproachistochoosexandfindthevectorb=Axcorrespondingtothatsolution.ThecomponentsofxarejustthecoefficientsinalinearcombinationofcolumnsofA.ThesystemoflinearequationsAx=bissolvableexactlywhenbisavectorintheco

6、lumnspaceofA.1ForourexamplematrixA,whatcanwesayaboutthecolumnspaceofA?ArethecolumnsofAindependent?Inotherwords,doeseachcolumncontributesomethingnewtothesubspace?ThethirdcolumnofAisthesumofthefirsttwocolumns,sodoesnotaddanythingtothesubspace.ThecolumnspaceofourmatrixAisatwodimensionalsubspaceofR

7、4.NullspaceofA��x1ThenullspaceofamatrixAisthecollectionofallsolutionsx=x2tothex3equationAx=0.ThecolumnspaceofthematrixinourexamplewasasubspaceofR4.ThenullspaceofAisasubspaceofR3.Toseethatit’savectorspace,checkthatanysumormultipleofsolut