MATH 218 Final 08, Spring 2017

AMERICAN UNIVERSITY OF BEIRUT Math 218 Elemeatary Lincar Alyebra — Section 9 Fall 2008-2009 FINAL EXAM, January 24 Duration: 2 hours Fustrociur: . Makhoul-Karam Solution Student name ACE 11 | Exercise | Qutol| GRADE hi 2 Fi Il 16 a4 TOTAL 10 snr Pt | i | he | rk | mt INSTRUCTIONS Ss hee Simple scientific calculators are allowed. Programmable caleolatars are NOT allowed. Write your answers oo the question sheet. Partial solution or partial justification will reccive partial credit. Khe back of ench page can be used if needed, and there i one cxtrs blank sheet st the end, Whenever extra space is used, alvays indicate clearly where the arader should contioue reading. Exercise 1: 12 points «= (6 points + 6 points} Zu 21 a) Verify that the vectors vy = G. oes she w= (22.- =) and vs = (=? =, >) form an orthonormal basis a4 +S for R* with the Euclidean inner product. b}' Express the vector w = (1.1/1) a5 lmear combination of v,, vz and ry. ce ae Evwlunte: the Evel dean ae’, Pduch *. Ml ge spwascusn a att _ z2 2 me a f-5 Hk a Thas , eg wee fers. [ment 28 2 2° etre ertho a Cane pape “Hing <Ainsrarhy, ob, Z 20 Sesee aa 1 fe Nee S| 4 3 pian thot (ot Val of an is : ea Cua [mn be tthe natn, eer. ear oe ined For IRE Woil= [Re Eek = ened - Simailorty | Yo bela les. Three Sore us aa erhenocene «bow St ‘ aber rhe frock ele. “Powak mph chaer a 4 See fom Uh, vsh AA es looms. for. ik, ath. mf; a4) cen fee sone ina ue tte fee TEE S Bawroe toimbainadion ef OF ub pe hak + en eit C2 he: Oy j i ayy, ry Ren: (1) a EN al \ 4 ny 4 Wa . Ty 13 Auf. & po mane a, art carfuhton, of sing wager. aye tes Cee BReetparipg=1.. Bc esste teens eee ~4/e.4 ‘3 fa wee tg =4. ee cette settee tee APS Ove ge, + BR cy 4. ok & SES Solve. sth oe uting (on ss - Veedlan elisrmeinahin' ay fe MB AQ fA A Mey tk ya 4S 4B ee l-3g fy ts | t ty -23 8 va as tS Tae J i 3p. (: ‘ Me. " gc? | : ‘ yeh Ye} : 0 4e Bayo fa 1 od ae 2 |— sla 4 © Sy \ aEAS oats tages ,C,2Y8 , 2 Sa wae eg [aeons 3 as method ogg {ules <4,W7u, + Saye, + Ce 17a) U3 vy a hy Fei an orthorarmad fea Yo yo ssn {vu}, - Largs in Exercie2: 12 paints G points + 9 paints) 1 @ a3 aa=(3 -1)ant= ZI. " io 3 a) 1s the hnear system Ax = b consistant? b) Find the orthogonal projection af b onte the column space of A ah Redacn He aes fenar keel amex xtieciated te As sk che 2b” Yua-eclw lon im : 4 a fl 4°99 4 4.4 4} rr" Ga fa}e fe le ]o eda). %, z je Pye aaly Ae oS pny fae entoy Vee Boe, dh .deowgae. art Eom, Ja trea sibel. bi Find the fast aQueares ols hon, of Aw= i 2 : “Thic is the exact net wm te AA = Wb (meskge’) ‘actors a keel! “tr Fina RA i - G 5) a oo. eg be ; t ae woe ae se a 4 ql ; ; f i en mR Fond AT BE ee (2) 4 C4 4) 49), Sowe Haws Gs (! ey =(") “2h o@ Ce Sys. Cas t=. eee a eae i om fh aaa | Thesefece , (o)-( 82) ae : - Raaat aaa ne eka Wen. af Pre ota | $ Orthagonal yaechion. af b orto The cafuimen . space Ww af Rees. gaye ($8) Pe Sake = 2 aC _ 3h “_ Exercise3: 12 points {8 points + 4 points} Consider the matrix A = ee ae : a): Find a matrix P that diaponalizes A, and determine PAP , 2/3 2/3 by Prove that 4" tends to limiting matrix 4, = (Gs 1/3 as m -+ 00, a} Ford Hr sifcnveduns of A, astubow of “tus Charackent he, efuahrn det [aT-A\eo es |*- G8 =a \=e a> N_agA af 20 -# 4 aa "i 5 AL 40 and odo +f ExQen uectors comes pencil. rte da 1! —— — Man ere aafn honk of ~ {at ck Ay Gao (= oe i aS (3) uo 9 ape Ea roychem. Bs aryivalut fe Hee bg eee ON Wns han 2Ruatton t sop tien ee & = y= E (aei bean peek. BL atten vectors are af Necers (2) ="(7) _ tet “vie (4) - x Etgeaue chors core eponeh ine —h Ae -O2 ‘ 7 Ge Ree we fad PAYA) ae Ste ere doebien tte, =O wy=b, Han x= oe EL ident bark Pye of Hi Sonim, (eet) EF for romain’ (go (%) * The mee ton £= (x |} =(7 : we Phage Mat. af thane. vor 2 wtochepee hae watenvicipss gl ia@fonatd A a cach orig iC o _ C a \ { diafenal..wuthic suns ; . OAL, o At ee pag effendolues bp We know dhat BARS BD es. A= PBT. % I mn r But Tv =(¢ coro pha oo. - o 02} = \e (ox Since | fier fo 0|, ane dedas Hhat D gats te. tg a steak al om =F, Bo “Teoeks “to —Raewahnt- motin SPDT] vee Fah A] od We Cf . 4 04 asenre(? “Wi OG a= GG 3 Exercised: 12 points o 1 Find a 3.x 3 matrix A that has cigenvalues A = 0, 1 and — 1 with corresponding sianvecors( i) ( -1) -1 i 0 and (:) respectively. lL {1 @ “The treats iar (1 Tac) = (‘, ‘| thes emerdet 2f 3 Dihaschs mit claw cate nver. toes eerie foo SZ athwct 2 vont) Ate. Conatigen SA ( ‘ A : aoe That 4s PTAP <D when D- Bae 2\(0 05) T nes g..g.. al ye Fired yp" ' boy rperfrin ing eon To she Sere row operons, ee Tookice A ; oh ts bet edits coed feuds Ges ze x 6 4 4 {. 8.0. F pA ese 4 |/a 4 0.8 4 mui 1 a o { oj\d a oo — -\ 4 4 ao oa 4A ~! di ata a°4 4 -1d off % - 4 9» of f & rh (. { 9 \ a oo (3 4 oy4 a 4 © 8 4 ]0 % o oa awe ¥z ae wipe De a a — Exercise 3: 12 points (2 points + % points) Consider the palynomials py = 1 +2 and py =1—x, a) Show that the set of polynomials fp,,p.) form a basis for Py. bh Let L be the linear nensipnaigeon from Py to B34, such that iL+x)=[ ‘ (1 Sy ange - x= C iy Find £3 + 2). “ ps0 polaromneedl vector equahen’ 8, Pt Bh =o. AVE @ (ax) +o, (-2) = 0. ay yCo wit clin he G0) + fe, ot, x = i a 2 tt Fellows en oa =o aack “Hears om Got, ao. Cee ores Hence tao fiw A Bouts pene “e (pany - te Bresickes , Rowing 4 “eat alton (2) = a = cnraatitye Fin } ane conclioes Hao .. An Pa wk al) ey tit Kos P, ble wale pe tte aes ene eeoneudhee of fat v= RT, + Ra Te : (see 4B eA aes 2) TH Slows! vay Ste = hy (Hoo +k, (toh. ae i Gs, 34% = (k, ae + (Rye ia ve Dems Has Aste aes kok, <3 hy Ry 4 i WE, hem and bed wba: p= 2c 4 Pal ey Ste = dex) + + {t- esi Fi Bor L 4 oo eel Tass Poeaatiown. | ery he ea eB) bist ante eel hax) Lissa (9) + ; 4) [Lees = Sl “Exercise 62 1épeints (3 points + 14 points} .f1 0 0 Ceusider the matrix ; A= ( Zz 0) dees ay Justify that the matrix A has a QR-decamposition. b} Find the matrices 0 and B ofthis decomposition. al dak (8) = 6 SS product af nme os ori da'afore (* Han (ular) det fe\o =) the cette ef A ae oa leres 4 tndepradanl, Tt Hotinuss ther A beac a CAR - cla compol ts we = by £ ae (AAs) , ural02,2) and ay (0,035... Sx Weote,tsy a. vce lopecs Kee the. coded fe w oy ray - ae : { Her . Wek?) Be ff % ort S Aa ar orthogonal lwasit | eeint Gra Schmidt pce evet G2 42 (440 and Wo span fad oth Her t2V3 e who ue Pry ke ae sige Me U = {0,2,2) - a) \ Ite Me = ($55.8) cotk fall (EF = Be ~ a hat Now WL= Spare ee Uri cgicangenny a e a = QR uakere Re (“ Os AL ¥ an] a) <M2,427 aus .4,9 y 7 e uw nef “Me <M 437 —SIRSf nae BNE Exercise T = 24 points {4 points for each question} a) Find twa distinct non-zero vectors w and vin R®, such that < u,p = 0 for every weighted inner proaduet on RA vfok url aay, eafO ay ouch Habs [Used , OHO, up and... Foo, oe F ca, UU \+ oC 8 TOY £5 thas From as idea ha, speoperte t SG acd uth=d, gees: (uzo tL use) and (u,=0 on #,=0) Lath A, uw, nek Stout La weoushy en fuad, a [sean genes Mertens ot Hus Roc rs uaa urlb cith ato, bto reatiehuy ee Ries . ( Exeenpes Lom cfhoh, v=(o0) ) b} Find two distinct non-zero vectors uw and v in R?, such that < a, % > O fer any inner product on R*. tet ta dR? foe can arte, tears hon —Fee arec hep... ! for any nner “product en TRE ea. het. ZuU,b5 = “u, ku =k qu ur Bee ‘ 5 - + td Crane | it (4,3 oe (8,6) 2 Bue bee c) Suppose that (vy. -..v,} is an osthonormal basis of areal inner product space V. Show thal for every w GK we have [felf? —< ue, Soe bee ue, Be, SF cies {oh nee Oak oan arthonenmad laaals , one fat bee Wuev, wt cu nr teu at : Uren } Bat ening ue ‘ to ae fuil*- Cu Wy = x Ly Sua, too $a, F ee = <W, cay fF -- GA, ue Ju, h eu t+. - - tu CU? = fume + 7-77 OF ae ie 4) Let a and v be veetors in an inner product space. Prove that ||w]l = [lv] ifand only ifu + v and w — p are orthogonal. ewta.t uj, lee artoc trary weebom , ia Se 4 dug, UP So. > Cue + ct ap bP eee Ve oe SO, es vjlel? — cues eure Mele eo ily (Pod? <0 & Niu jell. - Ce neta leeing at marckcs } a ee § 2 Pete nem revit) eunck and (he arn rhe & Aiagonals « oF he pova Uakefoa cavthnched ow a Deer find lat ee Thor birt: Me =e elit pe ae care “pe e) Let.t bea square matrix such that 4? = A. What can you say about the eigenvalues of 4? ofab Aloe an oi nub LAS ae Then Xsan an ovfenvedue “SPARE. But oA A rmnust thin aa 4 ne AeA GY AMUN =o me Ae ha, 4, Ay 2 a T} Show that the qpatrix, P= a we }. ovthogonal, TE wiz ay wee fC, and ¢, ‘ee dhe. coluw ptcters cai hyp Sere wk knee! eC .t,=0 anc < we oy ae. Peiter Follows shat 2 ie - erthafereh 2 \ 4 Definition: A proper orlhoronal suvriz ip an orthogonal mari: of whieh the determinant ix equal te 7. Check dat P is not — and deduce from Pa proper orthagenal matrix P’. dpe a relevant smell chonge oely) * eek (P= 5 ~ = 3 ale = mee & Fa eal ue Ce * Subectitehig a, ba Gel Rorps “Hla main orthogonal awed Chron fes vhs Giccwmnane ti, Se BY. (“ts a ) “Nik A