MATH 218 Final with Solution, Fall 2009

AMERICAN UNIVERSITY OF BEIRUT MATHEMATICS 218 FALL SEMESTER, 2009-2010 FINAL EXAMINATION Time: 120 minutes Date: January 27, 2010 Circle your section number in the table below: Instructors | Azar | Egeileh | El Khoury | Fuleihan | Karam | Nassif Section 8 5 1 6 | 4 2 Section 3 9 7 QUESTION GRADE 1 [25 2 [25 3 /25 4 [26 5 /30 6 /34 7 [35 TOTAL GRADE /200 Answer the following seven sets of questions; the back of pages may be used as scratch. : rE 2/4 1. Let T : R3 — R* be defined by: t 2a + y+ 3z L+2y T y = Zz (a) Show that T is a linear transformation. (8 points) (b) Show that T is one-to-one. (6 points) (c) What is the rank of the matrix representing T. (4 points) (d) Find a basis for the range of T. (7 points) ro cs (a) ae a _ 2 ~ * 4 ri _ 2 So 3 ‘ ONDE E yee PO Le 4! i e Ne 2 NM — R A 5 | 4s @ mmeotixn mul? pico how bu tha &XD matnrr TOs therefor a nemr trans ferme hon . On Tor al\ «=(\ and v=(y\ 0 R* , and Sor all ke, A ’ 2am 4 1 % _ \* re’) + (ys ky +kg’)\ ue Rv = Det RoC! ana Tle kv)= DONA CN 8 4°ny, * Grey) (G+RZ / \ 1 +h LENA , (ore d 4 aielalng Tas kv\= (es ak Gre — T()+ RT (v}. Hence the Linens * ’ 8 (b) Cina ama nvllspeee of Ts 21'0\ (0 Vvena) , vs (4) anak co mock Har TORO SS is Ts ; é a fellows a 20 bas =o which 4 #& hemo encavs f ainmat \ +2 =O Aycrern ha ving eo the tryin gr =e nolubhonw WE4RLRO =O There fore N (vr) ~(ol c. R = T UA ere -te- one (c) Reo @& ww-ecw low Foren f AY 21 3 3 a a) 21 5 a 3 : ay , 2 o\—> (0 32 -\— (0 % Yel |" : 2 eo \ t o 4 A p00, 2 . 34 e nee?) oo 4 o e Nt ad \ The rank of tho mater A ye presenhin +e Gineer draustorwe ee dy 4 eaual ~Ge commen dinmerso KN cak(A) and pws(AS « Bins the obtained row-echelon Brom bbs 3 Meackine eine’ One eoncQucdes that Terk (A)= 3S Cy Rerge fT1 ROY =cilAL — (T_ barng armaber multiplier” sae toh Sith gO eapahacem ox o! \ UA Rous -echrelou Gol 12-1 1 2LetA=| 2 4 -3 38 3.5 -6 —-5 (a) Find a basis for the N(A). (10 points) (b) Determine a basis for N(A)+ with respect to the Euclidean inner product. (7 points) (c) What is the dimension of N(A‘)? (8 points) Roew - echelow Boren ofA —t 3 t 2—-i ! A (0 ool i) (2 -\ -3 4)=8 o -) 3 -8 @ o -! aé A Axn=0 a ; 2 = a =0 bet sey = ES & a a ar nen 2e ae \ - Xs 4+, =O a, a 226 iN foo? 22 5 \ xen) & me(-K) 4 seldeng n= se NI , V legit, ie Re Re {a4 A ca lea Sit Sor nA. _—? ’ ce of &. (6) The acres terval com plemenh @f Ne) td “rss FOU? se L . . ae — an for NCA ts ao basis ef rows (AY - Sock tooo zould ke meade © ane That Wott ache lon Koren =f A con tevin’ ; +he deading Lanhes 4 \ : al OF) 9) Bis Quay uy ad ne re. “(| ee[\ a 3 (4 t €) sige CA) = 344 . Se (A) = Axo (wor) ae vank (M) = 3 Se “There re 4} a . y n \ Sone of But rank (PY) = sank (§) =3 (comans oh) peer fou) fy) Tt Eolouss ! um NAT =O - equivalently nyu Wars (©) =9 1 -L -1 3.Let A=| -1 1 -1 -1 =-1 1 (a) Find the eigenvalues of A. (10 points) (b) Find P that diagonalizes A. (10 points) (c) Deduce P~?AP. (5 points) (2) The agers wa luos f KR ano ae Reba Ho va =f tha. Charackon’ She emudton | (A AT) =0 [mA al al 4 a -t teA =t leo - rare DPN aeons tne aes Pou wy t HAL Fi 4iel ds) a- -AY TC -AP_t|+ La-t-T]- (1 4 1-al =e _A(\-d@- a) 42 (a-2)=0 GS (A-d[494- ay+al = ED (a-z)(-M+Aza) =O BD OAR Z =e —aé As? =o A=zZz Ac-t @D_A= (B) Finet _berces Ror the ergenspeces of At (then deduce r) ae Rignns Pats. jane ae +o a, HK (ef alaebrare wo lbabiy TS | Thies. wy Hae eeiirhben at ef (R- A,L)x=0 : eave a) | —t ( am =1\(&\= (2) { lt oe | Xz fo) . Rous — rechuaing +h ou wwentbesi quate t a are _-y o-\ je C3 ab)> 3 th) - = = a oe woes ae poate rays he nn eau i valent ta The yn. Lineal 22uation | x, 4A, ¢*%,=9 Ae oer os Cand « ‘aA d n a, = 4 x z a ae" cra ne 7 AEE ® Gy matetheety feriye' as eLALnS Ce Ce rreS Po rr Uh of A \ to a Ea and +a party ondapen ane ates, } Toy Ga bene dor Aa eaeepea. EiqenSpmce ch 2 a =o - \ [A-A,T)xco a ilo OT a = 3 |->( (2 32 “plo | ou ale 0 -¥, 2 6 - Beuryo ten wong r set 2x, ec Aa - £2 wy all> \ Sot ayet then et and apa t impQjing (lee Oi eens eat : \ 3h a rae hae Seer @-2 * -{ ab ot = % £=(y, [ai)=(+ 0 ) Ata fovalizes Pe 5, 9.8 2 9 oN (¢) 6 tt and P AP= (° Ay ak uv z o- oD A \ 4. (a) Without solving, find condition(s) on a and @ for the following system to have exactly one solution: (10 points) gtytaz=1 ct+y+Bz=2 arz+ By+z=-1 (b) For a = 8 = 0 find the least squares solution(s) of the linear system. (16 points) (2) The hi Ongar Zy3 Ags tern hes ca, UniQue shelulron 4 At he. coefircien ¢ epee ¥ wth PaAwer ts (Sey Theat ae Ag ay elatermrra Ak nom Yero 5 . 1+ tot eh 7 Cymer ol) @- A) HO |i i Rre Se (a-@] - eel #08 od & 4 E> so Gra ang we rt i aca ws ohiteusty dconerstenh, ) TAs colin fren ek | 1 « o\f*\i ff SS ae ‘ (‘ i >) ‘ = 2. iy 9 oar al Xx = WS ar mathe re A ns least SQuarks ocolubhean ~bo ~than Os : ma Zs the crack solution to ~he cores f 4 cnotenad artshem! ( Here “> r,t oO ! Bind AK: (\ I 0 and AY bs Parent 2 2 olf 2 2 of3 (2326) a ‘> (3 3 Ro stk Hence) rhe aiven PaaS tenn laas Nes Wk be pre ut A.Ros To 5. Let W = {p(x) € P3/ xp(2) — p(1) = 0} (a) Prove that W is a subspace of P3. (8 points) (b) Find a basis for W. (12 points) (c) Let T': P3 — P, be the linear transformation defined by: T(p(2)) = ep(2) — pd) Show that T is onto. (10 points) (a) WwW Sv los pace of £3 V p,qew , VRER Coal ion 2 ye Rafady- PM - Ra a = faye Reedy 7 = (grea) (rhe) wae ecaly + & [Hq —45 . mae \ — =o iva a hee Hence, (p+haye ALLEN ON hot W kk hosed ae addittn «nc Scalar trv bs Cas bio csacd Hasse ov loi Be Ce of fa: iy ~ Co) Basis Sor W; Ried fers yecbors, (qorynoenian % a VW pew » Pe ‘s of the ¢ ean plod Ae tS w+, came plas A+teAt art Fas (A) ez gtr A, An TA3 “of . oad “ep p =o. CEM =rotgrorsel) zo Wee®) raha (anna, 442s +a.) on fearak eee é $ ‘ ag . _ USING the tchanthy Peeper’. (er ta ea, "32,20 ; : ' ‘je \ t 4 + 10 (V2 aah) es & BG Ge tA tA, tA, FO = f a +3a,ttag ze Sette Age and a= A “Therefore , POM) Casex*) + a east gielae a> 238 Ae aud aa Ae + ae A % 3 = A(A-3E ee) + & (4-920 4%) Ler wtae2eBeaet and lye t-Fetoe - Then Woew, PF Ab + b, whert ate ® . Nees WN. «ea ih eee Bor _ eed ane Ounearlss tad pendent (distret che Crees) Therefore Jo,, bry @ leacnts Bor W : (c) Tes) = xplo} - 9) ¥ Pe Pg. 4g onto Netz gist thet W = N(T) Coal agence aT) ; wou We=lpGrse?, / Tq) =94 Rad diem (wh=d ytelding elton (NET) = 2 Knowing rok dann (én) + dim (Rr) = elim (Ps) ae lows! De dees(RO) =H GS dim(Rl=e. Buk RCT) ma aablspace 2 Po and dim(P) a2 soli ( Am Treffer RID can =, onto. ; 6. Let P) be the set of polynomials of degree less or equal than 2. Define the inner product on P, by: <P, q >= agby + aybi + age | where p(x) = ag + air + apex” and g(x) = bp + bia + box”. Let p(x) = 142, g(x) = 1+ 2a? and W = span {p, g} (a) Find \ such that (g — Ap) and p are orthogonal. (6 points) (b) Show that {p,q} is a basis for W. (4 points) (c) Use the Gram-Schmidt process to find an orthogonal basis for W. (10 points) (d) Find a basis for W+. (10 points) (e) Deduce an orthogonal basis for Pp. (4 points) > > 29 (a) (a -aAp) and 7 ae erthoaaral é=> iq-4p ; ES <q,er— AC P/P” =O . mite pie Sftzae | siete 4 A shovid then cooctis fy 4 4_-Qh=0 as A= he (bS Le ay bass foc We span pray Coe netcter the vechor <auahon! Ree +B.g =o Ave. Y2ceR, bk (A+>9 + h2(A+2x*) = 9 Chpeke) + hie + AR =O ae Follows i¢ +R. =? gun “thereRore b= Rr =O, Ry =e \ ake = ° Leis ew. rt a A the a ae om akan 9 <P 9 4. breasly do depends br and refore \ () Orthoqonad leasts Por W : Ye (b, sb) be Auch bess - 0 sans, Aine Gran - sbbonn fh LE pre ce SS £ “pet eb = P= qaoc 2 4A (4%) =4 Lala bo a $4 «b= G- Pega = 4~ Sa bi * ae a 1b, 3 b2| = { Aaoe Liberte | 1S an orthogonal leasts forw. (4) Bass Lor Wr = (fea4\e Broce {e-ay Oo qa walt rw y thon ! a\ = Es Va Y= ve wr & [eupr=e ane cui 4 to Pac ec eter , yieldang Z2yPV =H OTS _ LU, AY = Cot 2 So ve Ew >) Coty 0 = eg hey \ \ = be Ve tle = 2) ielding tue 2ayive bak rs ” ( 1 0 3 \t) “= (3 - als (oer rd, 426,22 Lok C, = Hyron C= 2b and c,=-Z2t .- ) \ ° So thet, ve W> GS ya —2642b% 44x" ab (zene), osl = {-Ue2erx"| AD @ leases gor Wt W ee) () ew and foybsy b oo Ther Th eth =o row =0 an? a,OC27— ) \ a Patowe hee A+% 5 4 dx +e p) A 2a does San ortrotral \ee-gits fe = 8/9 7. Prove the following statements: (a) T: V —V such that T is one-to-one, prove that if {vy,... Un} are linearly independent in V, then {T(v1), ... T(vn)} are linearly independent. (10 points) Con cicer she yechor 2Rue how ! cv ls) +--- 7 en 1CUn) = D T near => Tent ~--7 +S Un\ =e ») T do-one. ornc- a => Cur = oF 6,4, =O => e 6 2 Pu ond Oo ptedenl = C2 --- = a= — = only the dnuind TTharclore Tle) e-- 42, TW) =o hes only Teen ) ma k tog {lo ae TO Drsar ty pods pen chon fe. (b) Let V be an inner product space. Show that: |lu|| = ||ol| <=> u+v and u—v are orthogonal (10 points) ute enol uu ore ofthotonal E> uw u-Ur se LD eujwye HUT —SHITT mec ia 0 _ = & |lh+ 9 7 Z ce ulP =|ell (c) Let B be an n x n invertible matrix and T : Maxn + Mnxn be defined by T'(A) = AB. Show that T is an isomorphism. (15 points) % TT is abneat trons forme hen! YA AEM 5 VRER, T (Aska) = (Ark A.) 8 = A.3d +k A,B wz TA} eR TAL) Prenee the Ain-ve ok wT % Ts _anento-one Let NUT) be tte nullsqace ofT, Y Aencry, ThA =0 CE AB=O bub Bus taverhMe | so ABBSO (yo -mahn) NOW= {0% WreaLlGEnE TO ena-to -one % Ta ante: YM EMA. > if tere exitts a wats he auch hale tun ABSEM) => f= MB TOS\=" , Binverbel nines . Every matsxr PP) has & Ye tre (2 An Ainge wold wy A=nMeR “Th are Flore TT un an tSormorphiim: 3