MIT18_06SCF11_Ses1.13Graphs, Networks, Incidence Matrices

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1、AnoverviewofkeyideasThisisanoverviewoflinearalgebragivenatthestartofacourseonthemathematicsofengineering.Linearalgebraprogressesfromvectorstomatricestosubspaces.VectorsWhatdoyoudowithvectors?Takecombinations.Wecanmultiplyvectorsbyscalars,add,andsubtract.Givenvectorsu,vandwwecanformthe

2、linearcombinationx1u+x2v+x3w=b.AnexampleinR3wouldbe:⎡⎤⎡⎤⎡⎤100u=⎣−1⎦,v=⎣1⎦,w=⎣0⎦.0−11Thecollectionofallmultiplesofuformsalinethroughtheorigin.Thecollectionofallmultiplesofvformsanotherline.Thecollectionofallcombinationsofuandvformsaplane.Takingallcombinationsofsomevectorscreatesasubspa

3、ce.Wecouldcontinuelikethis,orwecanuseamatrixtoaddinallmultiplesofw.MatricesCreateamatrixAwithvectorsu,vandwinitscolumns:⎡⎤100A=⎣−110⎦.0−11Theproduct:⎡⎤⎡⎤⎡⎤100x1x1Ax=⎣−110⎦⎣x2⎦=⎣−x1+x2⎦0−11x3−x2+x3equalsthesumx1u+x2v+x3w=b.Theproductofamatrixandavectorisacombinationofthecolumnsofthemat

4、rix.(ThisparticularmatrixAisadifferencematrixbecausethecomponentsofAxaredifferencesofthecomponentsofthatvector.)Whenwesayx1u+x2v+x3w=bwe’rethinkingaboutmultiplyingnumbersbyvectors;whenwesayAx=bwe’rethinkingaboutmultiplyingamatrix(whosecolumnsareu,vandw)bythenumbers.Thecalculationsaret

5、hesame,butourperspectivehaschanged.1Foranyinputvectorx,theoutputoftheoperation“multiplicationbyA”issomevectorb:⎡⎤⎡⎤11A⎣4⎦=⎣3⎦.95Adeeperquestionistostartwithavectorbandask“forwhatvectorsxdoesAx=b?”Inourexample,thismeanssolvingthreeequationsinthreeunknowns.Solving:⎡⎤⎡⎤⎡⎤⎡⎤100x1x1b1Ax=⎣−

6、110⎦⎣x2⎦=⎣x2−x1⎦=⎣b2⎦0−11x3x3−x2b3isequivalenttosolving:x1=b1x2−x1=b2x3−x2=b3.Weseethatx1=b1andsox2mustequalb1+b2.Invectorform,thesolutionis:⎡⎤⎡⎤x1b1⎣x2⎦=⎣b1+b2⎦.x3b1+b2+b3Butthisjustsays:⎡⎤⎡⎤100b1x=⎣110⎦⎣b2⎦,111b3orx=A−1b.IfthematrixAisinvertible,wecanmultiplyonbothsidesbyA−1tofindthe

7、uniquesolutionxtoAx=b.WemightsaythatArepresentsatransformx→bthathas�an�inversetransform��b→x.00Inparticular,ifb=0thenx=0.00Thesecondexamplehasthesamecolumnsuandvandreplacescolumnvectorw:⎡⎤10−1C=⎣−110⎦.0−11Then:⎡⎤⎡⎤⎡⎤10−1x1x1−x3Cx=⎣−110⎦⎣x2⎦=⎣x2−x1⎦0−11x3x3−x2andoursystemofthreeequatio

8、nsint