MATH 218 Final Collection Finals with Solution, Spring 2017

AMERICAN UNIVERSITY OF BEIRUT Math 218 - Fall 09-10 Instructor; N. Makhoul-Karam = - Brief Quiz3 - January 7 Name:..... 8 alu.ben..... aay AUB ID number: ..........2..2.....02.068 Grade: 16 Question 1: (4 points) T: Pz > P2 isa linear operator defined by: T(p(x)) = p(x) + p’(x)- a) Is‘l one-to-one? b) Is T onto? ¢) Is {T (1), T(x), F(x?) a basis for Pz? tot : 1,2, | d) Find the matrix representation of T relatively to the standard6asis of P2. ET ¥ er £NCT) , pi)zaxtbaic sabRec TC p03) =.0. # Toswomial Ge. 2). +7) = a. WreR. . &. (ax? + bute) + (Qaxt he =O. , VxeR | &..ac® + (b+da)=. aes (e+e Nee 25 )» Vzer eo — asd 7 : to byda SO | _ Wielding ‘azbs “a route momiik.. is. the. only qackue. un Nan , aed elo. cpa NC Se oh i [ay one-te- ore.. bY TA) = 4.p0.=4. Ts St & tl. on, WHELs £ my Axe arenes wena | . C2oukdar. the. vtchor eauaten |. aa “belynacat = 4, TOY ee Te) +e TCA). = é, + ey (x+i}. + €3 (x + 255 = _ Week a; +) + (ende)y rz 20. YeeR CE rez =o 4+ 2 ield C2 £0 O ve Ps . wie aN the tnvted Aelufion ge ft 1; Te ee Liens ie en clant and! \8|= alin (R= 3 ie Sin a tae BrP Ay: 2, —P, Sine. Bz A as a lonnit or = rire YT as (2), ae is. a leans Lo Ct) c&, veld dung. Atm (RO) = 3 = sasen( and “es bis There\ore . 4s anto , 4 tre edialre .) weet TO) ed oe Ths) = eel aned Tada x*+ 2x. -fey HR) Ctrl) ee Be foc Bien Be all y Question 2: (2 points) Are M,, and P; isomorphic? Justify and find an mei fromM,,,2 onto Po: Neti. eetcthet.., Uy... gana. ion (2) sia & ane &,. aR the- ater tat +e {Rt iss 1 OW ed there Bore “teomergnic socal PX Adcerpa TAO YR er BAi ben From Mog ente i Ceyn la - a ‘Mea. OSINg the coordinate. rmope, ulabeety to 4, 2d Nouad-rasas Belen, fe, my rt! pe ” Oud... Bi {4 ae ye ae _ “fer E a RES x Het .anu.. difined oS Kollows ¢ ; ea peeas bxace4 dx Ts Grol AA ah ihommor parses a Ceompesiiton of 2 iSamor Bhiitwt) Nameticcnneinnenecsrens senpetes AUB ID namber: ........ CE CO TT) Grade: 16 Question: {6 points) : BaY 1-3 3 taatemsdone [3 -5 ) ae 2 646 4 Find a matrix P that diagonalizes A, and determine D = PlaP. (Homework continuation: Show that Vk< Z*, A* = PD‘ P™ and deduce a) see ric Atrrrerme Birra Bryon ate. saa sansassananeuaacnsenmnsaterersarensnsencaresoseprearerse eG ice Borde Gh sake \f CAH S\CA at) op BY. BL a8 (Act Bal) ERA aD) 3(-BAnb) CCA i2).2 sre web Red MAD ch CORD. BRAD Occ cccccceccsevnevessesseteeeccasneeessen PUD. 3) 2.0... ‘ AL..owe ecm... wekors >... (ATA). se axenusen andependent... vectors... comtapondnt...to dmad..deaund Sarin severe Bes baaShS. For. Ake. rores crmdng...eigen paca.» ee came: Enigenveckors.....catmsmonshing.. wha ASU sola. OOD IE Re... CIR ES . veaesee DT peng an seater. sagen GLa AN 2 a Pensey nurses veep: B» Seapeereers fate wei ye dye (ea oan) (San cae oen) ie Discs cece Spr eeOiinnee AE co RORY Boe NMOS S,... ee tOSG....1D ZZ... ventueeseess th the. opel. ergentaluel..0s itt... .uctetin've...duafeual. raunanesasenresii enytees.....- Luster. dn the, acedler..af?.sraatc..ecgen yeckne) hans obep.5. Teme. for hed. ( PUAP SB. 60... APS PD. od, Ae PPP} FO ns eee 7 ee OO tlo te \3 (2 4 #9 agit ee py ae ae OO sn fai wade: caper eens { 0 i! e AMERICAN UNIVERSITY OF BEIRUT Math 218 - = Kall 09-10 Instructor: N. Makhoul-Karam = - Brief Quiz2 - = December 17 Hamers: RBS oc AUB ID number: ........0...00000000 Grade: 16 Question 1; (3 points} a} Let T:R? = R? be the transformation defined by: 76) = (e mat where a,b,c,d © R. Justify that T is linear, b) Fora given # x » matrix M, prove that the function 7: M,, -> M,,, defined by T( 4) — AM + M4 isa Jinear transformation. al No that TCD) = (CES) @ (5) ad c, T se a omdtix omy IH phic how chuck tvare fore. a ater . F trac Let nahn . ar Dy previes the ft Rincas hy pren pees . > pz At x “Ee . ky" Vv uve R ) VRER | Les () We Cy) achat T lurk) ae (ytby’) c(t bx Se d(geky!) - {A+ DY | | 'ebuy? =.( Bee Courts) = THe) ~ R Tr) é T (Ar RB) = (ark8)M + M (A+R = AM+ R8N4 HA+ RMIZ | = AM4MA +®&(BMNIMS = TAS + &T(BS bh VABEM,, Whe® Question 2; (3 points} Let 7: R? ~ R® bea linear operator defined by T(e,) = (7) Te) = ( 1 \ an Tey) = (2) where fey, €2, &3} is the standard basis for R?. * 7 3) in the range of T? b} Find the dimension of NC) and R(T). a} 1s the vector w = ( a w is an RET Gif) thei exter. WS (4 ; Auch thak “TMs eS TG 446432) = G " % ow ES =(}+9()+3(2}= pat { 4} D ( g) ry OPES = (0 a2 9g dealt iw ul sy ( | ot 21S) {-2 ¢ -2 a | foal é 2) {a} ola | 215 y\o ot ate (0 2 2/4 (2 2 2 | Trcousis fenh ‘0 ~2 2 -3/ cA] o Ol -~l “There Pe me «ad da act an RCTS bj ners =f) 7G)=24 ares (3 enct) e 7 Bos: $68 989) = my yee ” & me TVET Utes +27 (eh = 6 a SON T Bt) i t(5)) , ax | i aim NOD] = 1 ¥ But Lier (NES) bir (Res) = chin (IR) = 3 a Arial ® (ty = oo.