Lecture 2: Elimination with matrices Video Lectures Linear Algebra Mathematics Download this transcript - PDF (English - US)2timesthisequation,Okay. Thisisit. Thesecondlectureinlinearalgebra,and. I'veputbelowmymaintopicsfortoday. Iputrightthereasystemofequationsthat'sgoingtobeourexampletoworkwith. Butwhatarewegoingtodowithit?

We'regoingtosolveit. Andthemethodofsolutionwillnotbedeterminants. Determinantsaresomethingthatwillcomelater. Watch Last Shift Download more. Themethodwe'lluseiscalledelimination. Andit'sthewayeverysoftwarepackagesolvesequations.

Andelimination,well,ifitsucceeds,itgetstheanswer. Andnormallyitdoessucceed. Ifthematrix. Athat'scomingintothatsystemisagoodmatrix,and. Ithinkthisoneis,theneliminationwillwork. We'llgettheanswerinanefficientway. Butwhydon'twe,aslongaswe'resortofseeinghoweliminationworks- -it'salwaysgoodtoaskhowcoulditfail?

Soatthesametime,we'llseehoweliminationdecideswhetherthematrixisagoodoneorhasproblems. Thentocompletetheanswer,there'sanobviousstepofbacksubstitution. Infact,theideaofeliminationis- -youwouldhavethoughtofit,right? Imean. Gaussthoughtofitbeforewedid,butonlybecausehewasbornearlier. It'sanaturalidea..

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Okay,andyou'veseentheidea. Butnow,thepartthat. Iwanttoshowyouiseliminationexpressedinmatrixlanguage,becausethewholecourse- -allthekeyideasgetexpressedasmatrixoperations,notaswords. Andoneoftheoperations,ofcourse,thatwe'llmeetishowdowemultiplymatricesandwhy? Okay,sothere'sasystemofequations.

Threeequationsandthreeunknowns. Andthere'sthematrix,thethreebythreematrix- -sothisisthesystem. Ax=b. Thisisoursystemtosolve,Axequal- -andtheright- handsideisthatvector.

Okay. Now,when. Idescribeelimination- -itgetstobeapaintokeepwritingtheequalsignsandtheplusesandsoon. It'sthatmatrixthattotallymatters. Everythingisinthatmatrix.

Butbehinditisthoseequations. Sowhatdoeseliminationdo? What'sthefirststepofelimination?

Weacceptthefirstequation,it'sokay. I'mgoingtomultiplythatequationbytherightnumber,therightmultiplierand. I'mgoingtosubtractitfromthesecondequation.

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Withwhatpurpose? Sothatwilldecidewhatthemultipliershouldbe. Ourpurposeistoknockoutthexpartofequationtwo.

Soourpurposeistoeliminatex. Sowhatdo. Imultiply- -andagain,I'lldoitwiththismatrix,because. Icandoitshort. What'sthemultiplierhere? Whatdo. Imultiply- -equationoneandsubtract. Notice. I'msayingthatwordsubtract. I'dliketosticktothatconvention.

I'lldoasubtraction. Firstofallthisisthekeynumberthat.

I'mstartingwith. Andthat'scalledthepivot. I'llputaboxarounditandwriteitsnamedown. That'sthefirstpivot. Thefirstpivot. Okay. So. I'mgoingtouse- -that'ssortoflikethekeynumberinthatequation. Andnowwhat'sthemultiplier?

So. I'mgoingto- -myfirstrowwon'tchange,that'sthepivotrow. But. I'mgoingtouseit- -andnow,finally,letmeaskyouwhatthemultiplieris. Yes? 3timesthatfirstequationwillknockoutthat.

Okay. Sowhatwillitleave? Sothemultiplieris. Thatwasourpurpose. Andthisguydidn'tchange. Nowthenextstep- -thisisforwardeliminationandthat. Okay. step'scompleted.

Oh,well,youcouldsaywaitaminute,whatabouttherighthandside? Shall. Icarry- -theright- handsidegetscarriedalong. Actually. Mat. Labfinishesupwiththeleftsidebefore- -andthenjustgoesbacktodotherightside. Maybe. I'llbe. Mat. Labforamomentanddothat. Okay. I'mleavingaroomforacolumnofb,theright- handside. But. I'llfillitinlater.

Okay. Nowthenextstepofeliminationiswhat? Well,strictlyspeaking.. Icleanedupwaslikethe. So. Igota. 0inthe.

I'lluse. 2,1astheindexofthatstep. Thenextstepshouldbetofinishthecolumnandgeta. Sothenextstepisreallythe. Butofcourse,Ialreadyhave. Okay. Sothemultiplieris.

Itake. 0ofthisequationawayfromthisoneand. I'mallset. So. Iwon'trepeatthat,buttherewasasteptherewhich,Mat. Labwouldhavetolook- -itwouldlookatthisnumberand,dothatstep,unlessyoutolditinadvancethatitwas. Okay. Nowwhat? Nowwecanseethesecondpivot,whichiswhat?

Thesecondpivot- -see,we'veeliminated- -xisnowgonefromthisequation,right? We'redowntotwoequationsinyandz. Andsonow. Ijustdoitagain. Like,everything'sverycursiveatthis- -thisislike- -suchabasicalgorithmandyou'veseenit,butcarrymethroughonelaststep. Sothisisstillthefirstpivot. Nowthesecondpivotisthisguy,whohasappearedthere. Andwhat'sthemultiplier,theappropriatemultipliernow?

Andwhat'smypurpose? Isittowipeoutthe. Thiswasthe. 2,1step. Andnow. I'mgoingtotakethe. Sothisallstaysthesame,1.

Now. I'musingthispivot,sowhat'sthemultiplier? Soit's. 0,0andisita. Yeah,Iguessit'sa. Because. Ihaveaonethereand.

I'msubtractingtwiceoftwicethis,so. Ithinkit'sa. 5there. There'sthethirdpivot.

Soletmeputaboxaroundallthreepivots. Istherea- -oh,did. Ijustinventanegativeone? I'msorrythatthetapecan't,correctthataseasilyas. Ican. Okay. Thankyouverymuch.

Yougetan. Ainthecoursenow. Isthatcorrect? Isitcorrectnow? Okay. Sothethreepivotsarethere- -Iknowrightawayalotaboutthismatrix. Thiseliminationstepfrom. A- -thismatrix. I'mgoingtocall. U. Uforuppertriangular.

Sothewholepurposeofeliminationwastogetfrom. Ato. U. And,literally,that'sthemostcommoncalculationinscientificcomputing. Andpeoplethinkofhowcould. Idothatfaster? Becauseit'samajor,majorthing. Butwe'redoingitthestraightforwardway. Wefoundthreepivots,andbytheway,Ididn'tsaythis,pivotscan'tbe.

Idon'taccept. 0asapivot. And. Ididn'tget. 0. Sothismatrixisgreat.

Itgavemethreepivots,Ididn'thavetodoanythingspecial,Ijustfollowedtherulesand,andthepivotsare. Bytheway,justbecause. Ialwaysanticipatestufffromalaterday,if. Iwantedtoknowthedeterminantofthismatrix- -which.

Ineverdowanttoknow,but. Iwouldjustmultiplythepivots. Thedeterminantis. Soeventhingslikethedeterminantarehere. Okay. Now- -oh,letmetalkaboutfailureforamoment,andthen- -andthencomebacktosuccess.

Howcouldthishavefailed? Howcould- -byfail,Imeantocomeupwiththreepivots. Imean,thereareacoupleofpoints.

Iwouldhavealreadybeenintroubleifthisveryfirstnumberherewas. Ifitwasa. 0there- -supposethathadbeena. Xsinthatequation- -firstequation. Doesthatmean. Ican'tsolvetheproblem? Doesthatmean. Iquit?

No. Whatdo. Ido? Iswitchrows. Iexchangerows. Soincaseofa. Iwillnotsay. 0pivot. Iwillneverbeheardtoutterthosewords,0pivot. Butifthere'sa. 0inthepivotposition,maybe. Icansaythat,Iwouldtrytoexchangeforalowerequationandgetaproperpivotupthere.

Okay. Now,forexample,thissecondpivotcameouttwo. Couldithavecomeout.

What- -actually,if. Ichangethat. 8alittlebit,Iwouldhavegotalittletrouble.

Whatshould. Ichangethat. Irunintotrouble? A6. Ifthathadbeena. 6,thenthiswouldhavebeen.

Icouldn'thaveusedthatasthepivot. But. Icouldhaveexchangedagain. Inthiscase. Inthiscase,becausewhencan. Igetoutoftrouble? Icangetoutoftroubleifthere'sanon- 0belowthistroublesome.

Andthereishere. So. Iwouldbeokayinthiscase. Ifthiswasa. 6,Iwouldsurvivebyarowexchange. Now- -ofcourse,itmighthavehappenedthat.