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(UM "l, a 53‘4"“ 06/ ('I'm LAM as” \V‘ “HAL 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 humanmaﬁa 55m always has inﬁnitely man]; antut'uana] (1:) The system always i135 at. least. um? aﬂEuﬂnu. no matter what In and n. are. _,
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(DU)5 wl‘Hq WWI? 1‘5) lathe? .a+ Vwbe' _. 4. Let It! hue3. m K n. ntatrix. and mneider the Finear hﬂmﬂgeneeus system ef equetiens ﬂfi' = I].
{Heme the mateLien means that :E' is an n. dimensinnei 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 snlutEnn. cell it
E. Em take an]: DEEI. number a. E}? the previuus suhpart. Eta—5' Es else a snlutEnn.
Thus the 53% has inﬁnitely may solutions.
{1'1} Give an example Elf a Finest: humugeneﬂns system of equetinns 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‘}.
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‘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 matting5 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.
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 Spring '08
 lee
 Math

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