solutionsoldexam2

solutionsoldexam2 - University of University of Waterloo...

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Unformatted text preview: University of University of Waterloo Waterloo Final Examination Math 1 I5 Fall Term 2005 - 1 Name (Print): , (Famil y name} (Givenname) UW Student ID Number: Circle your instructor’s name and section number Instructor Section Class Time Instructor Section Class Time B. Ferguson 001 8:30-9:20TTh 9:30-10:20M F. Dunbar 006 l:30s2:20‘l‘ 12:30-]:20WTh R. Moosa 002 10:30-11:20M 12:30~1:20WF F. Dunbar 007 8:30-9:201‘ 9:30-10:20MF B. Ferguson 003 8:30—9:20MW 9:30-10:20F W. Kuo 008 10:30-1 1:20MF 9:30-10:20W P. Wood 004 2:30-3:20MWF W. Kuo 009 8:30—9:20MWF P. Wood 005 3:30-4:20MWF R. Andre 010 1:30-2:20MWF Date of Exam: December 14. 2005 Time Period: Start time: 9:00 am End Time: l2:00 pm Duration of Exam: 3 hours Number of Exam Pages: [3 (including this cover sheet) Instructions 1. Write your name, signature, and ID number at the top of this page. Please circle your instructor’s name and your section number up above. 2. Answer the questions in the spaces provided, using the backs of pages for overflow or rough work. 3. Show all your work required to obtain your answers. 4. NO CALCULATORS PERMITTED. Marking Scheme: Mark Awarded Mark Awarded - 15! Math 115 Final Exam Page 2 of 13 1. Find the general solution to the system 591 "-332 +x3—x4 =2 2.761 — 232+333 7 5.71:4: 3 33:1 — 3z2+4x3 — 6:134: 5 'I fix t ‘l 1 ‘ Z —2 3 ‘5 3. ‘Qz'Zfif 0 ~' 6 3 ’3 L! (9 5 £35K) I w~J +I-l Z RN23 I o o (*3 ”' O 0 a R‘sfflz O O o O _ 025} 12:5 .791! F’fi W 3C, : 3,215+3 3&3: 319"! SKI SA2J7+3 .F: 2’ : 5 )6, I; 3'13"! X7 “13’ I _2 9 I. 0 if) 54’ 3 “t I‘ Name: C) -—O 0/ OZ) "‘3 «3 O Math 115 Final Exam Page 3 of 13 Name: 1 2 2 —i I 2 2 —1 . _ l6] 2. (a) Are the vectors{(3 4), (2 1), (3 _1), (2 4)}linearlyIndelmzndem'? Explain. . I 2 g 2 I 2. ] O ; —-( Z ‘1 3 Z Fl Z O M); g 2 3 Z 5 2; 3 O [f I —f L} A," l' “I k3 arc; \ :3 l 2‘! ext/LUZ. WIZMmmj 2.“! 1 / 3 2 3 ' 12] (b) Do the vectors in (a) span Mk2? Explain. 1% 11% OLD M Spam M11 «4%? {AI/:0 Math 1715 Final Exam Page 4 of 13 Name: 3_m=( I6] (:1) Find a basis for the row space of A. - Q)” flea}? I I 1 Z EVEN)” 22-09 a @0919, 3 l 5 gr g wr—‘N: mm 0 1 1 alt—tub- Do [31__ 03) Find p(A), she rank of A and pm), the nullin of A. (QM [/0 5 Wk; [flaw/24)) = Z W (/4) z #130 deal/morgzl '- W654) {L If f 2 : 2 Méth _llS Final Exam Page 5 of 13 Name: 4-wl={(f)=(3)}anwa={(W13)}betw°basesW [6| (a) Find TBHB2 the transition matrix (or change of basis matrix) from Bl to B2. Find [5132. i3] (mu-{5:13:42 3]B. l Math 115 Final Exam Page 6 of 13 Name: |8| 5. LetH : {10(1) :51-2 +bzv+c a+b+c=0}. Provethat Hisasubspace ofIPg. /- 0(1): 0150240 e H MW WOW :0 2L p,/:z)/€ob(x)é/7l w]? fl, (3C) ;“ 0 13+be 1+ Cr / (M33920 2 FL : (Ellipéle-F CL) aljygzfirclra 001- (1) ert = [OIMLDXZ+(LI+EZ)X+CI+CZ CW1 (OJ +01) 4' (bi +Lyz>+ [C£+CL> 1- (Cifibfg) +(024-bL—FCL) :. O + O I; O g 982% (at) wad ké/Z‘ W f, (79) 2 02,135,146” arkéfl-C, :0 i/ : {allvngbl'CJ — AW 1): fall/1+khl—bficl \ fiA why 1 ,meq) fiafi J : Jew) . :0 jgpfi) 9H fIL V23 QWQ@%/g_ . j_f71(})+001(1) €471 Name: Math 115 Final Exam Page 7 of 13 6. (a) Letz: 1—11 (i) Write z in polar form. a??? m £§E1€ZL 94 2’:J2:Mfl ‘f |2| I2! (ii) Find 2.“. 8 p A 29: J? 04: 5 2"”m‘owfi‘ 27[mwl%?+uAfi4Hfi) 2*(H'0i) .n ,- [2] (b) Describe (geometrically and algebraically) the set of all complex numbers such that fl = 2. 02,0631 2‘: : CH’JD L ~ I W :— a # 13 L. cud 2 : .L 2- : 1 (1*b C1+A L. :7 apb; I 0" Z + E L E 7‘ % C9; 01 ’ 19 L .2 a ,— b L ‘ aa-2 @351 G9 QZJEL‘ Wd ‘BS';EL1 aL-lrbt a It}? 2 _ (:3 03* )9 : ' Math 115 Final Exam Page 8 of 13 Name: 2 0 1 [6| 7. (a) LetA= 2 A1 1 . Find the eigenvalues of A. 2 0 3 1 A r >3: \ 2 O w» [ (2 WM) " 23 :0 (4%) l g, ,§/\+/\2- 2:? :0 (4700\ 1—5) “LL/J ’0 (» x ’A) (A - ‘00“) :0 a WA»~I/,A“~‘f//\:/ :51... (b) Is A diagonalizable? Explain. Ifso, find adiagonal matrix D and a matrix c such that A : CDC—1. (NOTE: It is not necessary to compute 0—1.) k 400 o~l'/ WD: 010 MC: }"/zz/o: QO-(IL 0 Ir Math 1.15 Flnai Exam Page 9 of 13 Name: 2 1 8. LetA=(1 2). [4! (3) Verify that ( i and ( 11 ) are both eigenvectors of A. :4] (b) Find A10“. _ 949714Ca) D:- 30 Md PZ/J 5] J _ a I I .1 Pi; ;' film 3/1] Wm A=PDP7 2, *I [j [A A I“: “0—1” “J' "000 V2 La] So A» PD? - { 1M5 fifth a. ;- SianJ _]/_L )1] [V3100] [fl/1.11]; E3] Page 10 of IS Name: Math 115 Final Exam a 1 —1 9. Fmd a]! values a. (if any) such that the matrix A z 0 2 —a is invertible. 3 1 —1 \. Age/J WW0 Math £15 Final Exam Page 11 of 13 Name: 10. Suppose A and B are two 11 x n matrices. Prove that MAB) 2 21(B). (HINT: Use the definition of null space.) mm W WMCG7 9* WWOW 02?} f G Wan/37 6i: 0 ‘ & wank/M & fl, WW”? A M] flfaszzfio :0 if; a Wag/AD - W653; WWW?) ~ OW mwm é wwmwwfl' Math {15 Final Exam Page 12 of 13 Name: .. 11. True or False. Briefly justify your answer. (a) If an n x 71 matrix A is diagonalizable, then A has exactly n distinct eigenvectors. Ill {,1 a - ‘ 3m/ H .7 m. flame-«fl? pMQZQpW .. '7 ,6! c Lng l M (b) If A and B are both invertible matrices, then A + B is also invertible. I2I 9M . wig l] W 31/1414] ml 1 l 13):! GM PFJFB"; 0 Mid [M Magi/5% is (c) If det(A) = 0 then zero Mn eigenvalue of A. [2! ’flZU é MM) 2 O e4” cal—i W033) 0 . a (d) if B = {51,172.63} is an orthonormal set of vectors in R3. then B must beabasis.ém [2! $5 $40 Mil/(04%.? fit/{m flawo; W ‘ H a 0215“ g 3 «Midajdlwmaof’ MW Vim a uecZEv 55mg.) J) Ou/IWW‘UW 3 aid/CA) H’Uj‘Ca% J Mk. 0a 773,06 . Math 115 Final Exam Page 13 of 13 Name: BLANK i’AGE ...
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solutionsoldexam2 - University of University of Waterloo...

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