MATH 218 Final Collection Finals with Solution, Spring 2017

AMERICAN UNIVERSITY OF BEIRUY Math 218 - Spring 09 Enstructor; MN. Makboul-Karam - Sectinn j Brief Quie3, - April 9 lunes gi ai AUTIE OD pamber: oo... es eseeer ieee Grade: ft Question 1: {1 point) Sketch the unil circle in R?, using the tamer product defined by: ene Juy¥, + Se, Y, . cotak.. ie = foe 4h. whee ont ‘Retnts a me Sy a STUDENTS a tT? AT WORK a oe r ‘Question - Find a ba oer a . sab. 3) " AA (16.2), Teplice: AL yyw: feb We ate BA. ge the. matin ela Se. mot ares boat co ee ed j crwemeiets ccatheesy hy ye ee pea Ca. AB. faeothrns le nce Depanee te oe rea. ete Tene ce, ae. A. aun. Has... a ee ae -erthageed.- Senate aie BE oo. -pearpacks whe Hae... Caseboesn es Lean pre chunch ‘he Wh atin. ae leo hit fr ae. rela a Ay ar dodanie ge. Pee =o... oe creat FA te hE noes chun ere, dou Ie aA 3 4 uk "a _ (5 eB Ne TO & -4 —4 i i re rs age agp ot Rape ope Deg fe ey inh 3} ox we elgg Ga “Bee * : =u +33 so . = a <a Vite eee _ o. 5 rt ‘ \ wet. HEE Y. sdd& 7 che S ig\ - . | Wecher fre Ee Bhs %, = CY aps - aN * estion 3: (1.5 point} Check that the set 3 = {u,,1;}, where uy = (113 anday = {0.2}, is a basis for &* Let S be the standard basis for R? and let x = (¢, 6}, Write the wansition matrix from the basis HE to the basis 5 and deduce the coordinate vetlor for p retative to the basis B. : “Travtition matix from Bin Sg (tH, | S41.) = (; y a STUDENTS mY. 7_Al WORK Show tha eh Bleed. Uys Up pry de mu, — See yy, AL ecalpag alk Haat), een - ue |] = ioe wy tpta vse. ,.. ilo Tite. ected o = (A, 8,0) eo: AMERICAN UNIVERSITY OF BEIRUT Instructor: N. Mfakhoul-Karam = - Section 1 INam@ieI 220s Sie AUB OP gumber: 20... Question 1: (1 point) “ Let May have the inner product defined by: (UF) = ir(QP F} = ech and let a(t 2 & : ‘] Ohwestion 2: (1.5 point} Verity that the vectors ¥, — (E. “al = pg) ice 3°33 33°63 Math 218 - Spring 09 Brief Quiz, - April? Grade: th 1 -3f tof liz ys" 3) form an orthonormal basis fir R? with the Euclidean inner product, then find the enordinate vector of w = (—1,0,2) with cespect to that basis. ea = Taree seeps a a a Chk Oe ~ Yenmraiertt f/ eat ey ex's) ce eee a S(-Heee 1 FF $ 5 a a Oe M5 cpunents My AT WORK —_ , Ba Bis ae eng: yug Qeestion: (2 points} WR YR 1A Lot 7 Let A= [: | and b | , Find a least square solution of the linear system Ax = b, and deduce the -1 2, 3 orthogonal projection of b onto the collin space of A. Is this least square solution unique’? Exyslain. x Tiret sobyee “thet Ha. Austen. Pywsla ud J atinconet hear |b t byt ‘ j » ne Hence i thu, Baereir. east ems fe bldaet pe epact aatiite, Fa lq Bey foe Fe. Ga S6 BSG BR Ae. Go AMERICAN UNIVERSITY OF BEIRUT Mach 218 - Spring 09 Fsstreetor: N. Makhoul-Karam - Section 12 BrefQuiz3¢ - April $ Ee AUB ID gamer: soo, Grade: 6 westion d= (TS point}. — Determine if each of the following matrices is orthogonal. In the case it is, find its inverse. o1 Ye ha Le gs A= : let yj he once i wwe, STUDENTS | — = laid AT WOnK mm Tee omatex R lacie elec columns east. a . earls on a. vara = Br = (ng WE WE YE Vik. we/ Question 2: (2 points} xty4z=0 Let WF be the line in RR? defined by the intersection of the pwa Planes: a Ko-ytzeif bs me Find parametric equations for 7, and then an equation fir rt. o i. CC ‘ mA \ tf ° e, Br ‘ i Q o : 4 i te oo te re en E. iss RieKa z, ‘ ef 4 eof =e. Ph ors vrreaest eA =E- segue Fe (lous. shee, omfg: pee Ine... PP (- Le 505 5 etal sto Han - ene AM eat Heveefere = Sane imant chs waft a Question 3: (2.5 points) Let R? have the Euclidean inner product and let 2, = (0.0), 2, = @,7,-2), 2, =(h2}. WO EAE ane eA “kh = = (0 (eh. sve wantin to ue Sooo cen setegent tems Be RE worm in _ Las tih sl Asa a AMERICAN UNIVERSITY OF BEIRUT Math218 - = Spring 1 _ Instructor: M_ MakhoulKaram - Section 12-7 BriefQoniz4 - May 2h BeMmetsicie UL sree Sop AUB ID sumber: -....-.2..-..0 0600002. 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Et wget wah + Seggnaeston ioe ae < he.. engen. oar I PE eating othe orflalane es siagbe ne. tf Gt a BY Oe. fee ee bec ee Ae Hee ee cvcesseaes oe Go bee eeeinlah agin pm ore ag or. — “hh wets... mg EE... a | ep WW AMERICAN UNIVERSITY OF BEIRUT Math 213 - Spring 19 _ Instructor: NM. MakhoulKaram - Section 12-7 Brief Qniz4 - May 2h Bammer sii es caeee AUB ID sumber: -....-.2..-..0 0600002. Graile: {6 (Question: {6 points) LetA be the 3x3 matriz: 4=|/35 -3 3 =~) STUDENTS 7 AT WORK he MA, seattle PAV stetliccc Loat Panne fA. {. eer) eae Oe ee a +2) Santa uae a. — nding. siQenvectns creel. lnceter. for 2h pene. Par enbtcin rs. ied para] FES penthng.. ce anBaos zs Alt. neniigen. eee ha fain, 2. A AD ee. LAL Axe a ee ween A Pagid, op en ol Aaa. eS ten. . eee enqaiunlee t oe. eae ee ee ceeeeees wandependent. eigen ccbors.. orth Bonin i Rp hen SOF iv, we, Lye. _ for. fe, RAMETERS. “eugene ce. ei he sober ~. + Seggnueston ae ae Re ete nee «eee PE eating othe orflalane sagen... 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