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1st - CHAPTER 2 Systems of Linear Equations EKEHGISESETZJI...

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Unformatted text preview: CHAPTER 2 Systems of Linear Equations EKEHGISESETZJI 1D. (3.} z=3wlflt,y=t1whem-oofilcoo. {h}:1=3-3a+12:t,mg=s,:c3=t,whm—oo{s.t{oo. [o] :1=r132:5,:tg=t.:4-2fl-—4r-23-3t.whoro-oo<:r,a,t{oo. {d} U=h-W=33E=—¢1-tn+5ia-7h.y=latz=¢m‘fl'hfim ~W€3ht2l¢ag hqoo. 26. {a} BisobtainedfiomAbyintorohnngingthefimtmdthlrdmm. Aiauhtninodfi'omflhy interChHIg’ingthEfimtnndthirdm. {b} BhobtfinedfimnAbymfifiplyingthnthirdmwhyfi.Aiaohtninodh'omflhymulflplytng thnthirdmwbjr i 31.91] 3cg+oz+2cg— {24—5 Dz+3o3+2ci=fl -C1+fl2 +5¢4=5 2c1+¢2+263 =5 DB. [21} 'Ir-no.chorenrenzflcolutnnsfihenthofimtna—loolumnnoononpondtothoooofioientn olthnmriobloothatappenrinthooquntlonsondthohotoolumnoormpondntothemmtm that appenronthoright-handsidonlthnmualaign. {b} Mao. Hofmhgtoflxmnple 6: Thomunnoooflinnnrayotom appearing in tho [Eli-hand column nl] ham: the some solution not, but the wounding augmented matrioen appearing in the right-hand column are all dlflemnt. {4:} Palm. Moltiphringnrnw of the augmntod matrixbyoerommoponm tomnltiplying both aid-esoftho corresponding oquntionbgr men. But thlniaequivnlent tn discardingonn of the equations! (d) Thin.chearstmnismnoiotont,onnoansolvefoctwnofthominblnaintormofthathirtl orflfl‘nrtherrodnndmcyinprmnt]farunaol‘thmvnrinhluintermsofthoothertm. loony om.thernisatlnastnnn “Eros"vnfiahlothntcnnhomndnintonpnmmotarindmdbing the solution set of the swim. Thus if the 31m in oomlialitolmtI it will have infinitelyr many solutions. EIEHDIfiE SET 2.! 11. The given matrix corresponds to the system 31-53: +325=-2 £3 +ltrn= T n+5¢5= '3 where theequeflanmrreapmdingtntheeemmhnnbeenomltmd. Sohdngtheaeaquntinnn For the leading variables {$1.23. and ml} in firmer-{the free variahlm {an and $5.} reunite in xlzui+fiwg—3:rs, agar-£115 ends. =E—5m5. Thamigzfingarhitmyvelueetnmg and rhthe salutionset can be represented bythe parametric equations :1 =—2+fie—-3t, mgr-e, zazT—4t, sures-fit, 35:: where —ea<:e,t<:ee. Theoerreapnndingveetorfinrmin (11,1333, 54,175) = [—2. I], T, 3, El} + $5,], 0,1}.0} +t{—3. 11-4. -5. I) an. The augmented matrix ofthe new is 3 2 —l —15 5 3 2 I} 3 1 3 11 fi —4 ‘2 3D and theredueedmwechelenfinrmnfthiemnrrixie 1 D I] I} I] 1 I] {II I} U 1 I} i} I] I} 1 Thus the system is inconsistent. 45- The ahIla-neuter! mntrlx enha system n l 1 T —T 2 3 IT -16 1 3 a“ + 1 3a This redrawn to 1 l T —T «D I :t —2 n a {13—9 3n+9 Thelnntmwenrraepondntn£n2-9)z=3n+9. Ifn=m3thieheenmenfl=flmnd theme-temwill have infinitely many nolntiene. Ife-S. then the teat waterway-end! to {1:13; the system in inconsistent. Hnfiiithenz=fiifl=figmifmmhnekaublfltuflnmymdzareunlquely determinednnwellgthenyetemhaeexerthrnneeelnfinn. D8. [3} The. FerexnmpleLE flmheredueedtneither Eilnr [111 (1". {b} Fahe.Theredueedrnwechelenfinmofnmatrixleuntqnn. {c} MnThenppmneenfnmwnfmmnmthnttheremenmeredundnncyinthe system.Bmthemnainingeqnnthmmnyheinennsietent,hnwmeflynnemlutm,whm infinitelymnnarnelntiunn. filleftheeeere possible. . . nfthe d FflmfiTheremnyheredtmdnneymtheaynbethrmpleJheeystemmnflstmg {} eqnnfiomz+y-l,2e+£y=2,end3r+3y-3haeh1flnitehrmnnyeulntlm. FL {I} [fag-Emthnnthamdmthnmbeamplhhedaafifllwa: a b 1 5 1 g .5] 1 [I] [a d]"[c d]-'[n “592]" u 1 A u 1 Ifn=fl.thenbg£flmdcyéfl.mtheroducfiun¢anhewfioduutasflnflows: u 1 c 1 5 1 g [I a] [c all—“[11 3*[0 1]”[0 11" u 1 {b} If [:3 1:11.11Innarad-11111:!£111[“1l filthfifithemjflSMQPBrflmaflthBW matrix 35’; wmmdmitmamatmonhemfl 5' fl mdfinmthisithflmthat themtamhastheuniquamlufionx=ffly=h CHAPTER 3 Matrices and Matrix Algebra EIEFICISE SET 3.1 4. {all masize2x4,ATl:-assimedx3. (b) E"12=211IIF21=‘I ('3) 5i: = 1 ifanfl on]? if {i1j1' {2:2} 01' {2:3} {'5} HUT} '[1 4 3] {a} 0293?}: meu 2 22. {a} uTv=-[3 ~—4 55[T]=fi—23+U=J22 I] a a 21 u (11] mi": --1 [2 1- n]: —s —23 a 5 1n 3:. u {c} triuvTI=E-'EE+D=+2'2=11TV 3 {d} vTu=[2 1 n] --1 -fi—13+fl-—22=uTv 5 T {e} 111211115") ntrhru'r] =u-v =v'u-uT1r=1r 1.1:- —22 ee+ee eons 15+ 113 11 . . _. au+el some eu+25 e1 53 e5 34. {a} Theentneeet‘themetan+J- ”+9 “+12 45+“ 31 n, m repreeent the 15+e w+1u 35-1-11 23 to 4.1 total uniteeeldineeehetthe categories during the InmtheeflfhleyendJune. Fur example, thetetelnumberofmedimn reimeete enldweeMu-i-Jgg =4fl+ It]: 51}. 15 2'1 er: :1 t' 15 3 53 2.1 T 311 M June eelee in each of the eetegefiee. Note that June sales were [use than May eelee in each cage; thrue these entries represent decreeem {h} The eutriee of the matrix M -J - repreeent the difference between May and l 45 W 180 1 -. 311 m Ill 11:!) {e} Let: [:J.‘I11enthemmponentenfflx- 13 65 t5 [1 = 122 representthetotelnum— 15 4D 35 I 90 her {ell elem} nfehirte,jeene.euite,endretneeeteeoldinMey. {d} Let3r=ll 1 1 1]. Thentheeempenenteel' {15 Ill] T5 3D 3|] 4|] 1? 65 45 15 4C} 35 1PM=[1 1 1 1] =[1ee 195 2115] mmlhetfltHleEIDfsmnfl.medium1mdlargeitemsmldlnMay. 4.551115 1 11111 _ 311110111 11111 {e} The 91:1thny [1 1 1 1] 12 554,5 Hzfl 1 1 1] 1:! =493 representethe 1511135 1 911 hotel mmher il'iterne {efleleeeenrleetegorleeheld in May. 1:19.13] False.FnrezempleJt‘Aieitxfienchitx2,thenA.BendBAerehethdefined. {1:} True. lfABendBAerehr-thdefinedendfiismxmtheanuetbenxm;thueABie mxmnndBAienxrt.If.inaddition,flfi+3fliedefinedthenfl3mdfidmuetheme the same size. he. :11 == 11. [e] ‘I'rlne+ Ben:- the column rule1 1151:1113] = 1113(3). Thu! if B hes a when 01' meme. then .113 will have a column at" :eroe. (d1 Fame- Pm me. [3 1H? 3] - [i 11+ {13} nut. IfAienxn-t.thenATiemxn.AATienxn.endATAientxm. ThueATAend AAT hethereequare metrieee, endentrIIATJl} endtrHAT} ere hnthrlefined. [n Fl“.Humdvmlgnmmmfihenurvieenflxnmflm P21 Since Ax = m1e1{A}+ 3202b!) + - - . + :rnenLfl. the linear eyetem Ax - h is eqfivelent to nettle} + mental} + ~- +mnch} - b Thmtheeyetemdx=hieenneietentifendnnlyifthemterheanhe pressed ' mmhlnattenufthe column vectors of A. ex as e hneer ...
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