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math103hw1solns

math103hw1solns - loco a k’t 1‘/000>[I'I/Io;23Q b...

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Unformatted text preview: loco] ,{ a § k’t 1‘ /000 > [I 'I/Io ___';23Q' +, b : 7'90 ”51° I ?%O 4% [I —y,o\ /00’0 ' 7 logo 41% a O 10990—7[<>|rooo 5 A \HOO ? IHOO Hoo gouach— ‘Gao/ \IOOO \o= “>00 r’;(loo)+/oao: 980/ 3t» 160:): (L “9%) + Maw MO 4379+ 11%; 31% 3 '7 «94 (2J5) ' (r? (”1+Lib)ooo0,b) + Q540m€®= l?m(&t) ® (zu‘HO-R) mew) — (4«+L)MC&) =0 TL}, a?! W M (M AC \QW ALL VJuaaogij" #10 we, set 31 M: Q: (a) '5 ‘Fg/H (“D M “AL PM Vhfi‘ (Marc & deacon-AIM»; slope. 51' (0.131,) F? 'F'f (Q) 3 film) W M2, tut 56/ CW 0'6 ‘HNL “Opt NWT bL ob'scmplrinvolé Ut (”CL/b». 150W usurp» SW {ween waving fi’(«o>=o: Hm“) ms M roHCrCOMJ-Er 5% M W-(QA ax W9 Salim, \fiWa M (ml) Po'mlcs Can-Jog) 130,.” 5"VMJ' /9wk|-v~3 Far W ‘FUHLHM “EU-")‘Pn. Cn3(3q:) + Chm/(2%) +CIML :O' (UM "l, a 53‘4"“ 06/ ('I'm LAM as” \V‘ “HA-L UnkM Cf, IJ' 3. If fun have a 53.5mm DE :11 linear humogeneuus equatiuns in 1H. variabtes. than justify the Fullmxing: [a] If m c: 71, than at least :1 — m variahies can be chnaen arbitrarifiy. IUD] If m 4: T1, the 33mm b.3345 infinitely 1113.11}? flfllflEiflns. [This means that if 5.131.: hmre fewer equations than turiafiies: than the human-mafia 55m always has infinitely man]; antut'uana] (1:) The system always i135 at. least. um? aflEuflnu. no matter what In and n. are. _, @OU( Sjghm (a... LL wH‘H’U‘ fig ’47:? W\M’c (AF is an m X Y‘ ma“ x ' F“ JH‘LQ A Mk: (Te-F we, [AM/o k “M“W‘W‘ 6 M (DU-)5 wl‘Hq WWI? 1‘5) lathe? .a+ Vwbe' _. 4. Let It! hue-3. m K n. ntatrix. and mneider the Fin-ear hflmflgeneeus system ef equetiens flfi' = I]. {Heme the mate-Lien means that :E' is an n. dimensi-nnei teeter]. (E) If 11' E IR“ '3 a. saintien elf ELIE = fl. and n: E IR is any number. then shew that m? '3 else a Solutien. {Eh} I]: 11'. “F E R“ are tum eniutiens tn the system. then 511w that E + f!" is else EL eniutien. {12] What is wrong with the fullmlring Iugie: Hy question 3e. we knew the SFELBIII fluff = If? has at Least ene snlutEn-n. cell it E. Em take an]: DEEI. number a. E}? the previuus suhpart. Eta—5' Es else a snlutEn-n. Thus the 53% has infinitely may solutions. {1'1} Give an example Elf a Finest: humugeneflns system of equeti-nns with exactly one eniutinn. [IE 3.11: were not mnv'umed that the laugh: in the previous was fluent. then this should mnviuee you] @ 51km MH’X MN. is a Auteur orwt'm; MGGZ} : «(M377 3 «75a ‘6» @740“ 9;.“ ”hm WNW i: Mi: _,) MW”) = Mfi’ + M‘x7 : 3+0 = 0 ® ‘1; ow 5:"th Y7 IFS JaflL ‘03 +14% ~ (4-; . J7 - ~37 WLHJA \S V15" 9\- Mtln/ <0 V M 5. Let. ELI be 3.11 m x 11. matrix and E' E IE“. and mnstder the linear system HE equab'teus ELIE = E. Suppuae the pmpemy in queshiflus 43. and 4b hnhi: Eamety if E. 1" E [litn are two sniutinus EU ELIE: Ethen an E5 [ti and 13+ 17". The shew: that: the system must be humugeneflus (Le. E: {7‘}. [You can 35mm that the 33mm 11115 = Elana at least I:|1'|.E ankutian] "vSuwose, Wt IAN/1 ax SoIU'HO-A MIX:E?I'F 0C6? L5 pJSo IL (elu'l'viv 14F ”:7 délK)‘HN-VJ Mex?» : E? . gulf MON): MMUT=oKZ 59 (X? = ‘8 «pa( 0W3 délfi. In [whoa/4r ‘H'lij M95 7; 4:0) WM jvvw 5. Shaw that. E ynu perfurm mlumn eperatluns em a. matrix M". then Item will nut. change the raw rank. [H.LNT: Fits: Denim young}! that. This is true for matting-5 that are already in mm Bduabauu farm] —-a ‘FUFS’V '{J‘Jg prove. HMf {L M '3 I.“ (‘réla/ “HM/W WC CALM" betrm‘l4ran 096v?" 0W 'HM. rod rawh- WCU M W; Wit we, 4W“? 2. Colt/was. M raw {Mk LS” M, MUMH( 0'? ram/5 '\V\ TF6}? \Hfimf 11M W M w‘ rrbc WTVA SM"!— (“fa-5A. (Lufi Afif/ W" m é/JIJIM Mia 41; Mi 40" CL we, W WT Co‘vwn W “Al/:4: MFR iv~ (TO/"P Jed? MM NJ “yawn WM «4, obferve h/wT ”W COMMU‘\‘¢ lg V? WC, ‘4) k MW 116 rauJ Maw, in MI mum MAL"? 04A or column 09- Wk «f M IL rows or Och/mud) Ml aj‘ “Misfire. 1K9— SuLm-‘vrx ...
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