exam1fall2002

# Linear Algebra with Applications (3rd Edition)

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Unformatted text preview: Math 253. Full 2W2 Exam # l I. Trim ur Pulse 2. Thar: ctiﬁLﬁ Em immihle 3x3 matrix A such \$31.11 is lb: 2.2m man-1.1 [It _2 h. 111ame | isinmﬁbieforallreaJ munbersk. L n. [H is a 2 x2 mania: will: delay = 2 . than the system A} = must be mmistenl. a L 1 d. 'i'llm: :xiﬁlu 2x3 matrix A and: 3x2 mantis Ewell Hun. AH=I:. c. If Iht: mtn'x A Elm-mums with B. and ﬂ cummulw wilh C“. then A must enmme will: 11?. 2. Each of In: linear Irmrmﬁans 1'11 pans {nd lhrwgh [c] mpunds to on: [and anh- onclloflhc matings}! 1hmugl1J. Match them up. No- enplamlicm is neadﬂi. a. Scaling b.5hlnnr nkntaiinn iijminn tﬂcllmtiun H20 11!: 3:2 El E: 41¢ :13 9:? a; £le a n 1, 1 n, M 4115 it '1; |—; 1 PM 03‘ ._M {m‘ H_2 "I bra :- _n.a 4115‘ 'M "1.1.13 '93 03‘ ‘1 'I n _ﬂ.6 -n.a 3:. Find all the Ex 2 mlriucs Ihu intimate with Twamm'nr will cnnlain me arbitrary.r enns’eantt‘a 0 l 1:. Find a 2x2 manimfmnkl IllnlI::aI:nrnln1.1t¢7:s.wiThI2 ll. . {I 0 1 CI . 4.1“ pan-ts {nythrnuuh it). ﬁndn 21-: 2 man-ix J. “barman a a and I} I . WIth'Ii'bc given pmpurtr {or explain “by m such matrix axis-u}. 2 1 , I [I 'I ll— 3. a'=] 2 b.' A=iﬂ D n.A'=.=i dd: c.A-‘=. - 3 4 ‘3 6 in n u — 1 :n ' Mall! 25}. Fall 2W2 Enm # I, Salt-Hons In. F "'1 isim‘enible. then so is J!.ED‘I'J'IEI.1:A:=I=O. h. T The delcrminam is. k! + 4 , which is always positive. but mun: Ell. _ . 2 u. T Maui: A is: inumihie. and um {unique} sahuim orthe gym is x = .4 ' I 2 1nd Iﬂ d. T Examplcsare .42” I Band 3:0 I - an n:. F Let 3 = i 1. for example. and Icl A and E' be num- mwicea that fail In unmmute. #1} [it ilﬂ h. E Thcl'annataf'n venicnlshaa: man-inf: 5* I] 23. D The forlmt ﬂfﬂ. scaling nmm'x i5: _b' c. C Thefﬂrmatﬂfanﬂaliunmatrixis |,wlwre a‘+.'1"=l_ II {I {I . . d. A n I represenls- a premium onto “1:: vench axis. a. F Since a reﬂecﬂﬂn m Ira-13111 and angjcs. 111s: columns ufa reﬂection maniac must be perpendimﬂar uni: vac-Inn. ﬂfﬂm remaining only F isufﬂﬁsfbnn. h Wigwam 0 1 u b u I- U l" r 2-5 u—h c J m . = .u. = . . 2 -1c d c d 2 —15 2d t-d' zu-c zb-d 25:4: u-—.EII=d_. 2d=2a—£. c—d=.ﬁ—d. It is required that c— 115 and d—a—b (ﬂu: other two cquaﬁuns being ":ulumhun"}, am I} I that the I‘MI-‘io-IMMmuling wilh In 1L| are oE'thc font: a 1* 1b .ﬂ—h ‘. “hurl: (land I? an: ﬂl'h'illll'j' mama. M253. Fall ELM-3 h. A 23-: 1 matrix has rent I ifit is neither invertible {rank 2]: nur mm frank D]. Among In: the matricea we ﬂaw-Id in put {a}. we acct 111%: that m nonzero and have dﬂnrminlnt i}. NM: a b :d _ 2: _ = 1b Pb «:2 ml: 2:: Lt: 2b1u+m [I «l when n =3: or a: --Er. Thanh: miuii-uns are ofﬂlefurm 2b 1:- Ab Ina a . l b.’ my: 25 h :23 III-"Mr: 15 m. -1hll Ya“ were asked. to ﬁnd only an: afﬂwm marines (furl: speciﬁc .5}. I 4 Hi __ 1 _ 1—2} -3 1 . = a+-c b+2d ﬂ ﬂ:.Thjsmz-.ansll1ma+2r=ﬂ t: n" 3u+ll5r 35+|5d El 1!; and b+w=llan aa—Ic and h=-M.Thus1hemlmimmnfﬂ1efmm '3" 4%.mech a: mu. .9 d #2 l 43. A=[.4"J_l 31"}: —If2 c. Gains Iimugh our list DIWEMI mafnnnniim wc malizc 111m any 1 m I] “Elk: example. In 1. EH»: bga+u1g_ pmj-ectiun matrix {mm a line} will work. but A =| I“: New: u — 1 .—1 I“ {E d c d cia-i—d'l d‘+bcj D -1 havebun-H.1othud:=—l.whi:hiuimpussihlc.lfu-duﬂ.or{ll-a. 111m l: a! + 11c -. :1" +1»: .— — I. which in impossible. Thus an: n mam: lion nut axial. .H'a+d:=tl.|l1cnwu Going Waugh our IE5; ﬁfgc-umcu‘iml mafanualima. w: maﬁa: L113: u. mminn ﬁuuugh 21,93 has Ihis properly: —J§ cusIZ-IJ'JJ -- eLn[27rf3} ainﬂnﬂ] mama} 4’5 _1 ‘2 {'l'lwrc an: my nlhnr mlmians} ...
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