5.1(2d,3b,8,15,16,17,23) 5.2(3b,11,13,14,18)

5.1(2d,3b,8,15,16,17,23) 5.2(3b,11,13,14,18) - HWb Maw/7...

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Unformatted text preview: HWb Maw/7 11’3/ HW 8 So/mh‘ons jech'on 5.! Vri’ém), T(a+bx+cx‘) = (-qa+2b~2c)—(aa+3b+h)x +(aza.+b+5c)x7- fl: ELK-1", —I+x2‘, -l-x+x°‘} T(x~x") = ~‘{(«I--Jc:-ur-)o:") TC-l+x"): -24-“,0) ==> [T]p= 0 0 3 o —2. o T(vl-x+x’-): 3(x~x’~) -1, o 0 Noh‘ce. 09’”) & (~l~x+x."') are no+ el‘jenvec-Ivrs 50; i5 n01L a basis (onxrsv‘rnj of 3,:ienyec—farj of T. A: 0 ~z —3 «I I 4 z 2. 5 (i) Defer/nine all eigenvalues of A: cle+(A-A1'n)= dei- ~A —2. ~3 = —A[CI~A)(s—A)+z] —: l—A —« +2[(—:>(5—A)+z] 3 Z 2- 5‘” -3[~2 ~ch—A3] : ~A+é>x1~1m+é = *(A~I)(A-2)(A~3) x) As: ’, A2: 2., A3: 3 are, ayenvalues (.7) Elyenvecwrs : (H52 mm 5.9) Form-7!: N(A—I-I)= N -1 -2 ~ => -v,-2v,~3v3=o —: O —« -v, -v3 :0 Z Z ‘1 2v, 4»sz +1193 =0 .So/ufior»: : g f (:9 J' 1967 l .4 -2 -‘ -V‘ ~2Vz —V3 : 0 For A3=3: N(A—3'I)=N ‘3 -2-3 ._.> ~3v,~2vz~3v =0 2. z 2. 2v. +2Vz +2v3:o soluh‘onS: t ("‘) ,‘éeF o I (‘7‘) fl: 4 I “i , 4 51"”9- #hese #we teEnvecf‘DO‘ are linearlj I'ndeleendent. 61"ACQ—‘D <-"-‘> A&= 621) and one may check +11}; .43 doinj am“ +he ma'l‘n‘x mulhflt'rah‘on. @ Claim A linear aperahr T on a fi'n4‘+e~dxmen:fonal V€C+9r Space is inverh‘b/a :‘FF- zero is no+ an egenvaiue, a; 71 PF [€476 be a 5451's for Me Vecfvr‘ space on whrch T 1': ale—{1’11 ed. We‘ll jive a pf”)! 5} wnfmpoj’fh'va. ‘ 0 is an agent/glue of T <=> de+ (ETJfi ~0--L <==> def (1:17p) :0 (=9 [7'75 1‘: n of in Ver-h'bl 9. )so (=7 T ,3 “of invéf-Hb/C . T is :‘nverh'ble (=7 0 i: run“ an eigenvalue. of I. m (1,) Le+ T be. an inVEr-h‘b/e l/‘neAr opera-hr, (M .A Scalar A "J an eigenvalue 0;? T 65% A-‘ 5‘5 an age/wane, of T“. 1315 A i: an eigenvalue of 7‘“ <=) 5] vet-for v¢0 Lb. Tv:/\V cu» ‘7‘"(7’v) z 7"(Av); since 7” exm‘: <=~> v= Arm») ,- 53 It'nearf'f'j a; 7- <=’> A“v = T“(v) ,‘ since A95: 0 £3 <==> A" is an eigenvalue of T"_ m Hwfi 3 Le+ Tbe a linear operahr on a vec-Ivrsloace V, and :21- x be an a‘jgnvec-far of T coerSPonclin] 4v 4h! agent/glue A , Claim Tier any [perms/e. {/2115 er m. , x is an ElyeflveC-f‘pf‘ a; TM Correspondinj +12 +112 ea‘jenvaluc Am: [if Clearly flu: is flue for" m:/ : 7'00 =A K. Asxuma Tm(x)=;\mx‘ We'll AhDW 7""“(XJ z A’”"’)( 7"”“(»:): 71 7"”(u) : T(/\M(x)) =A"" TL’X) =AM-AK =AMHK . M . TM(K)-)§ x fit all pen-five. {ml-eyed rm... a @ Claim Similar mq+r£ce$ have +he same +race. [BE [2+ A and B be dim/Mr ma+rrces. So 3 I'nver-h’ble mq-fn'x & 5.6. B-‘GV'ACQ. 172cm, fr(B)—= fr (614/461) = fit ((A&) 6Q") by 56: 2.3 EXJB 3: fr (A), m Define #fie +mce of A il'near apar‘ahr' 7" an a fink/e. olxmenyrona/ vac-for spare V. Le+ ,5 be a bow? for V, szv'ne +he +race of T +0 5;: t,» (ETJIB) any 775,3 1‘: wellwlefi'ned since if fl “M4 J/firewv’wa 5M“ 7"” V Jhan [Tjfi and ET])/ are Simf/ar‘ by 72;”: 2.23 and 50 m 1:3 tr([7'3,e)" t’(£TJr)A @ (ompm‘a-h’onal anxwers.‘ @ fie efjenvechrs wrreipondz‘nj +0 AH are +116, nonzero Symme-I-n‘c m4+rl‘ce5. 71.2 eigenvedvrx LorfeSPOnda'nj 412 A24 are #15 noneem 5keW-Symme+fl‘6 ma+n‘ces. :: ‘0 00 O! 0! @13 Moo): 0.). @4303} @fi: {mug-sh} U2E,3z]$i<j$n} UEF‘U- Mam; gm} ‘ where D; is fig nxn diagonal mm‘n‘x WP”! I as #12 HR cll‘ajonal enf-ry and 0 e/J'éWhere.) 5,3 is #12 mm maM‘x mvfin I as #he .‘J'm emf-(y, ./ a5 flejifit enfry, and 0 anew/26d) and F9 ,‘5 {he nxn mm‘rl‘x wi-fh [as 501% Me. 56% 4nd. JIM en‘ln'fi and 0 ekewm‘ MW?) @C’lflj’m If ffié) {S #16 chumden‘S-h‘c Polynomial cf a dz‘agonalt‘iabie ll'hetlr 0/96th T: “"5” FUN-‘7} , fiae zero apera-I'vr‘. Pf Assmm’. T: V-> W ; V, W wed-w- Spaces. .—. 57nd T {5 a/I‘ajonalgable, :7 a basis for V conjz'xfinj of e/‘jen V€C7’vr5 of T ,- Mron i+ by fl:fv,,v2’...lvn} 7716/: if )(éV’ x:c,vl+...+cnvn for. fame Jug/ac, (numb). Wen fCT)(x)-‘ HT) CC,V,+-~ + (at/o) = (.FCT‘) V, + + Ca {(7‘) vn )- ay /,-,, gaff—{j =C.FCA.)v,+ +Cnf(/\0)V,, by , where /\; f5 #1? elfjenra/ué torrwpondx‘ng 1v 4%)? el‘jenl/eL-l'ar v- I : c, (an/Mu- + cALOM/ns sinne 4108 A; are, .me foo-IS 412 41a? Lhamdfirl‘lfic polynomial 155*) = 0 . 50 {(T)(x)=0 Vice V. ,. " z: t- ”A 0 ' _-: i”\ 2._ mam m de (0,” c x) (A z) I *A _- (I—A)(A+/)(/\-/) Since 441e, characfl’n‘fh‘c Poiyrmmfal {pl/4:, (mdfih‘an I of +he. +851“ far dt'aljonalizah'on i5 Sa-jq‘ffi.e&(" [TJF has eI‘jenValuES Afi-l WWW mulfifilr'd‘fj l A: z' I a n 2' WM; ‘04 ofA2=L 71n— Azu : s-rank(T+I) = 3 ‘rmk 10: = 3-2, =1: mwlfi’oh‘u'i‘y 02.0 O/A :_’ lo; a . 5o Candih‘on 2 is Ja-h‘J/recl. T {S dr‘agonaltyab/fl . = 20h?” lo, *1 ft(") +s(0’) ;s,éel&f )climCE,h)=2.. f 0 I . fl: ix,[+x", I—x‘} 1': A basis for V 5-6, [:ij rs oli‘ajanal. S“ p ‘ Z ’\ r-1 "l L—J ‘73. $ H ll 2 /—'\\ c, I Q o 0 Q .. V U @ Le+ A be an mm ma+rfx MM is similar #1) an upper fn‘tmjulqr main); anal has fie dl‘xh'nn‘ 2132/: Value: A, , AM...“ Ah “nun. (OHM/,ondmj mu/th‘L/‘I‘W'QS "1., mg)...’ Mk 1‘. @ Claim ff = Z MI‘AI oA=I ff Le+ B be. Hm upper huh/gum,» mam‘x 5.6. A :‘s :I‘mimr +0 13. 75m 53 5mm Ex. Ma. , £764): 17(3) . 8 - {:I Since 81‘: upper +n'anjulm- , So is BIMIWE and +hu5 Madly»sz mm, WW MWM‘Q w (55.4%“):va Aeé (E—tI) = (A.,~ t,)(£,_l—é,_) (AM- t") where 6;,- fir'e {416 dl‘ajana/ Cam's; of 5 and. +11! éfS are #412 at'flenvalues. We see fiat f;= 5;: 369/” each 1_ 5M“ find/n, ma-fn'cef I‘m/e +113 A"qu chnmcv‘m‘rh‘c [Mignon/7m! and Jim: #28 Sam? QIBEnVfl/HGS wifiw #2? Aame mH/fi’pll’d'h'és we, know each fi; {5 egua/ +0 Jame AJ- au+ of ANN/Ah. In finch 6,“ = Some A} fun— lsjsk. n k 713445 fr (A) : 6r (8) = 12;; = m.‘)\; m H: I: @@ Claim 6mm): (Mm (A1)“z Mk)“ ff Sinze similar ma+nzes have egual d€+€rm/'nan+5 and B is upper 1‘n‘an3u/ar, +1124 0126A = data = 5., Lu--~1,,,,, = (ADM‘QQM‘ (AQMK (“HIV-zinj our Work m @3. 221 ® @ Le+ A:(' 9 ‘ Tfien A*: (I 0) . Nome +ha+ Me 81:98nva/u25 of 60m 0 Z [2. A and At are I and 2. For A:l, EAUU is: anneal 6 I I” j 0 and €A(At) 15 Sfannéal Ly (—2). @@ 75"”? 50mm” = (myth c.5271) rcze—Zt(!) ‘ .— yl: 3x-y 9 3 (59 M If 7‘ and Hare “mu/famously oblajanahyaé/e, cinema/‘5’ .Hqgn T and H commode... (Le. Toqu‘). .— P’c 5ince T And (J are fimu/‘I'fififouf/j dl‘ajonah‘jaélc, Han Ordef‘é’d 555:3 l6 fur V if. [T]; 400‘ 6V8 507% dl'ajorm/ mafn‘ceS. 75m [Tl/Up = (TL; [01le ,‘ by Cora/lag 1‘1: 75,». 2.11 (p.89) 3 [uj’g I:ij ; Si‘ace dr‘ajona/mafn‘ce: commufé’. = [L‘ij TU: UT. m N015"? "f 2/13 any ofher M5): , {hen 3 a charge 0/ fovrdz‘nav‘w gnaw”?! P 3.4-. [Ty/3 = 1"”:ij l9 and [7.3” = P" [71y P and fram M15 we je J’ka'f" [Tufiy 5 [HFVV M WW. ...
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5.1(2d,3b,8,15,16,17,23) 5.2(3b,11,13,14,18) - HWb Maw/7...

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