线性代数期中试卷2011s.pdf

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1、UNIVERSITYOFWATERLOOMIDTERMSPRINGTERM2012StudentName(PrintLegibly)(familyname)(givenname)SignatureStudentIDNumberCOURSENUMBERMATH136COURSETITLELinearAlgebra1ForHonoursMathe-maticsDATEOFEXAMMonday,June4,2012TIMEPERIOD19:00-20:50DURATIONOFEXAM110minutesNUMBE

2、ROFEXAMPAGES(Includingthissheet)10INSTRUCTORS/SECTIONS001SeanSpeziale(11:30)002TingKeiPong(8:30)(pleaseindicateyoursection)003AndrewBeltaos(12:30)EXAMTYPEClosedBookADDITIONALMATERIALSALLOWEDNONE(NOCALCULATORS)Notes:MarkingScheme:1.Fillinyourname,IDnumbe

3、r,section,andsignthepaper.QuestionMarkOutofDon'twriteformulasonthis110page.2.Answerallquestionsinthe210spaceprovided.Thelastpage311isforroughwork.3.Checkthatthereare1049sheets.4.Yourgradewillbein-58 uencedbyhowclearly69youexpressyourideas,andhowwellyouorga

4、nize78yoursolutions.810Total75MATH136-MidtermSpringTerm2012Page2of102323331.Let~v=405and~w=435.00[1](a)Findjj~wjj.[3](b)Findtheanglebetweenthevectors~vand~w(youmaygiveyouranswerinradiansordegrees).[3](c)Find~v~w.[3](d)Findproj~w(~v).MATH136-MidtermSpringT

5、erm2012Page3of102.Considerthelinearsystemx12x2+x3=02x28x3=84x1+5x2+9x3=9[1](a)Writetheaugmentedmatrixforthesystem.[4](b)RowreducetheaugmentedmatrixintoReducedRowEchelonForm.Makesuretoindicatetheelementaryrowoperationsyouareusing.[2](c)Findthegeneralsol

6、utionofthesystem,ifitexists.[1](d)Whatistherankoftheaugmentedmatrix?[2](e)Findthegeneralsolutionofthecorrespondinghomogeneoussystem(thatis,theho-mogeneoussystemwiththesamecoecientsasthegivensystem).MATH136-MidtermSpringTerm2012Page4of103.LetPbetheplaneinR

7、3containingthepointsA=(0;1;1),B=(1;1;1),andC=(1;0;1).LetLbethelineperpendiculartoP,containingthepointA.LetU=(1;1;0).Findthefollowing:[4](a)ascalarequationfortheplaneP.[3](b)avectorequationforthelineL.[4](c)ThepointonthelineLwhichisclosesttothepointU.MATH

8、136-MidtermSpringTerm2012Page5of10[5]4.(a)GivensubspacesVandWofRn,denethesumofVandW,writtenasV+W,asthesetofallvectorsinRnthatcanbewrittenasthesumoftwovectors;oneinVandtheotherinW.Thatis,nV+W=f~x2Rj~x=~v+~wfo