exam1spring2002

Linear Algebra with Applications (3rd Edition)

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Unformatted text preview: Mat. :51. 5mm: Exmnfll 1. TI'IJEDI'FIIB'E a. Ifthe matrim A and Bcurnmutn. than matrices A! and l? must rmmmm: as wall. In. It! is any squat» matrix. than the itcrncl at' .«1’ must be a subset ofthc Iva-net am. e. Ifthmc matters 5.11. that it'- in R' are lineartytlcpcntlenl and A is a lit-hi matrix. then: I'll: venting AFFAF, and Ail- multt bcdepc-ndcnl nswcll. d. Ifal in 2x1 muixwitl: MM]: 'F__ mm flu:va muathc inlhc imagu of matrix A". u. Thu-n: exists an inwrtibte 2x2 matrix A mutt that A" r-ri . 2. .. _ _ 2 I 15 a. E-md tit: sunllng mat-rm A that mfnnns 3 mm I"al . 2 t] I}. Find that: pmjcction matrix B [1:31 [EMSfOCl‘lftfl- inlo . 3 E. Find thcmflmion matrix C tl'nl [maskiner 2 2 3 am. I a t :1. Find the rotaflnn maLrix D that hansl'urms 3 "J a. Firm! m: slum matrix E that Irmsfunm I 1 3 6 9 in all that parts offltis problem“. 4 6 J. LEI :1 — 3 '2 4:. Find m:|'t.-l'] . b. Find the rank of matrix A. c. It matrix A imniblc? d. Is thy: irmge ofmatrix if a lineman; .0: all ul‘ R“? E. 15th:: kernel Ill matrix A a. line. a plant. {E} . or all 01' R"? mm. Spring 2m: pl, I . . :1. Let A = ‘2 m burn pans ol'lJus problem. Shun-all your “Wk. 3. Find all mnlriocs 5 [Jul mule willl .4' . "r'uur mm mill Domain 5mm: and-limin- mustants. IJ. Find a matrix C will: all affine. {allowing pmpnrlits: - 11‘ wmmutes with A. I {7 fails :0 be lnuertiisle. I- f fails: In in: a scalar multiple of“. Thai entries of 1mm unswu'r musl lac spaciflc numbers {nu Erbium}- cunsmms plenum}. 2 1 5. L¢1 A = [3 4] in all 1h: poms ofthis pmhlcm. a. For which valuesufcunslafii’ rim matrix A — H: fail to be =F: 41*] b. Find a nor-mam vector in Eh: WI of 111:: matrix A — 5h. invertible? :. Find I. mer via-mar .it such that .11.? n SE. Ill Mum m. apr 2m: 3 Enln 1, Solutions Ln. 1' Awntawu 2:11.433 hamyqflmutmmdtmnml ' u u” . 0 “i=3. c. T If m? 1- MT 1- cf: E is a nontrivial relmilm. than 5:: is gliui' 1-H + m? r .45 .. u I 'l‘ . h. l-' Cumich J: D “1|,wilh 1611A}=spmlnl. willl kcr{d‘]=kcr or. Mil-MIT +wIfi'= E“. flail-inn that 1“men's AH. AFnIh-ll‘ mdcp-cndmt d. T H Lt-cltd}='.-' an II}. Ihcn A isinwniblmand m '15 A" .5u Ihal hu{A’}= RE. —I t. T Considcrimatiun 1hruugh mm. for example. wifll A “I? ID I 2. me a sketch in each case l I] " {I a. Scalingman'icasmnfflheihnn fl *‘:fi'fimt=3$$fli=; 3. . . II] I]? b. m: wanllhc pmjuctinnnmulhcwrflw ummmfl=fl 1‘. l 2. WI: 15ml th: mflcclinn about Harlin: 5mm:th I1.‘I.II.I'iil1ani’lIiJ: L‘= a}. . . —h . . .I -h 5 3 d. Roulhun mtnmmoflhcfom .wnh a‘+b‘. Wen“ f a u =14. fl 3 q .ura=fl.&b=fl.fi.mthat fl=| {3.6 413 [Ml H.45- ' e. We. mt a vet-lieu] shun. with a malrix uflhc Farm . II is Iquircd 11131 '| n13 _ L [.12 - I 3. a. rrefH}: n U a. J I o” al.5oihalk—1m'nd £42 A. 3 sl'w‘ 3k+2 :fil‘a‘r 3 o 1:. min-11:1 C_Nu:5i.l‘l.lll: mfg-11:1, u r: Ii. Mdlzspan 3 j isaiine a. term} isthe planewifla equaiim 1', +2.13 +31, =I'J'. mm 253. 5pm: 3m: 4 4. 1 WWW a 1:1 I] 1 1a 15 n+1}: a+2b1 gm b+d = .or. = . c d2 2| 2 3:: d c-i—Zd c+ld| 23+}: m+m This mnnunls M111:- cqlufiuns 1'!- -' a and n+5 - d . with general snlulion u 15 J3: .whmnmdfimuhl: eat-Imam. 2b u+b ""1" I: Wewanl L‘ a a ' put 11—- . '— ,a:.1n w: 24'! 5-9-5 EL' ._ u h : : WHI'C-lI—Mzh “+3: --r:I -I-ab—2h --l[a-2b][a b}—D.mflmlfl=—2fim a:b.Tu make sure lhntfi'im'cncatarmultiplcul'fl.mlctu=-3b.Fwe:-wmple. . #2 | JAE: b mmn=Lwehave£I=l l.Mnr¢gmerally.mymmfixC= 2b bl. with mum in. dues the job. 5. l-& 1 t n. chm dnl{Jnkf3}=d:l 4 k ={2._k14..k3u3=,e -5545 =(k-5Ik—l}=U.lhflli5.I=50rk=L —3 ' —3- I b. JI-Sf1= 3 ] .Tflfind Ital} l.mlwlhcnqualinn n3xl+x,=l].ur. I I I I I :3 =31]. Amnmu wlullm t: In 3l: anynmm scalar muluplc mks“ 1mull. c. Aizi't' mulhal AE—fi=fi.m.{A—Sfifizfi.Wemlmricingrmavwmr . _ . 4 1 _ 1nthekmlufd—3I,.nsmpmh.LcL x=[3i.flfiulpflrlb. 5 I =5 ISI [3 :53 'd :Im= ...
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exam1spring2002 - Mat. :51. 5mm: Exmnfll 1....

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