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solutionsoldexam1 - w ' SOLUWo/ur University of Waterloo...

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Unformatted text preview: w ' SOLUWo/ur University of Waterloo Final Exarrlination3' Math 115 Fall Term2006 Name (Print): , (Family name) (Given name) UW Student ID Number: ‘ Signature: 7 Circle your instructor’s name and section nuinber Instructor Section Class Time Instructor Seetion Class Time ’ B. Ferguson 001 8:30-9:20TTh 9:30-10:20F R. Andre 006 2:30MWF _ F. Dunbar 002 10:30-11:20MWF I. Moffatt 00:7 10:30-11:20TTh 9:30-i0:20M F. Dunbar 003 8:30-9:2DMWF . . W. Kuo IOQB 10:30—11:20MF 9:30-10:20W P. Wood '004 - 12:30-3z20T 1:30—2:30WTh W. Kuo 009 8:30-9:20MWF P. Wood 005 3:30-4:20TWT11 P. Wollan 010 l:30~2:20MWF Date ofExam: f December 1'5, 2006; Time Period: Start time: 9:00 anr End Time: II :30 pm Duration of Exam: ‘ - 2.5 hours Number of Exam Pages: 12 (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 PERMIITED. Marking Scheme: Math 115 Final Exam _ Page 2 of 12 Name: 1 2 UV 1 [7] 1. The matrix A = 2 4 1 .—1 row reduces [ORA = U 3 6 1 *3 {J 2 :| . Find a basis for OC‘JM GHQ 3; 9 (a) Null(A). ' ‘ [276 fiequ/W. Lei ELL-"=3 aflzf 54* 3.3.2.416 afiaéfiézj (c) com) (d) What is the rank of A? {61:11:54) 2 4.1 (Math 115 Final Exam Page 3 0f 12 Name: ‘ 1 2 1 2 A 2‘ LfitU: span { I , }. _ t1- 7 -1 4 (a) Findabasisfnr U, _ M I .1 i‘ J - I 0 “I g 3, 311:, “m 0! a : , L; 1,; L, _ 96, M3}? I £5) Cs :1} 9 (In) Find dim(U}, dIMKW ‘3; 3 .7 1 i “2:. g 1 s a, Math 115 Final Exam Page 4 of 12 Name: 14] 3. suppose will = 2, um; a 3 and 93.17: _4_ Find "356; m // 351 ~95”: (33: mag) .(3’51 mag) ‘1 ‘Nifi-fi) M "I9. '31 54,: 6x g “m1 “91%) WWII“ q ?"2’+‘/Y+§!«f ' =1 3% film 3%; ; we, a 5" WWW [4] 4. Lety':(1,1,1,1,1)andf=(1,—1,0,0,0).Let U={:EER5i:E-§=0 and f-i’:=0}. 'Prove that U is a subspacé of 1R5. 0:» w H: - Math 115 Final Exam Page 5 of 12 Name: [4] 5. Prove. that 0 is an eigenvalue of A if and only if det{A] = O. , Mg“; {J m/iL zfiz-w/A'éég; 4%.” Mwira . 95%;? (.4!wa 55» LAW: Méfli‘mfl} : th¢;' {1} M“ W "14/ éfi/fi) 7313 174“- cf, m»; M £94) Hug/“545,4 WW3 (—14:12? Math 115.FinflExm-n Page 6 of _ 12 Name: 1 O 1 - {7] 6. Let A 2 [U 1 D] . Find a diagonal matrix D and an orthogonal matrix Q such that A : QDQT. 1'0 1 ‘ c (w yaw a -r ‘ «g ’4 " Q at“! 9 "w; (59’) ' r!" L 6' .m-‘fla _ 57355 lb "'1' 0 ' ' 0 I g M [‘0 w; o a; "‘1 [019 o- A _ "1' 0-1 ‘ 0'63 a 9‘ ‘31 $37”; 34:"3 a”; 5{. \RIR" 00"! 0 [00 Q 0 if) Q o "7 Or)! 0 ""10 a‘b 999 5 31?”: ,3 3L, 22%;:3 m 5/10) if.) Math 115 Final Exam (3) Find a basis for W. Page 'i of 12 a Q'UL r931 30:; +3.1“? Jia “was "$0 <7 mm {Eu ‘3‘, 5:1”! Gila £3 0 1‘"? Wfi (’fi 0 p t "75,1- 25 3441:; / W! .V 5 f’ 5" 6 wt} ’ o '(b) Find an orthonormal basis for Ui. Mg {f A Whit i .1 "‘ ( DUNN/bang, [/ mfg > 6k :5 4m \ 'I’ Name: ) P l' .a ,1 '0L'(_12)é3w/§_ r, "tr if L H a 213 6‘4 «2 Eta-i“: Qlfajfifl> 35’; ‘7»? {P’s In 1’4 e 36am ) 'M- "a Math {[5 Finai Exam 7 Page 8 of 12 Name: [6] 3. Consider the vector space 1P2. (a) Are the vectors {:5 + 1, m2 + as, 3:2 — 2} linearly independent? Explain. 51’F3§g_ ' Q{g..§:} ér é<§3§fik%i fl; (3;?er 5% i =7 “QLM #633!" ’0'?!» +Ca‘4ré'la «m (fig%¢);fi%+(2:1i§)ag .+@,3¢ % W5 I ' lo iv; “5‘3 Q“ “r7” Cr “‘11? Tim: Lag aw mgé’wé (b) Is 1:3 -_|- 331: + 2 E span{a: + 1,352 + 32,5112 — 2}? Explain. Sifiti/ 515:5; 3%,1 [pr’mHQLIQLQ 33L; din; gar ‘ _ $16“ Km - QQ'MMQ- fiat/‘5;th {agsfwaimggw it 3’1 “VSJL Ml .2- af’m m) +5‘j‘gfim) MC»?ng % W96...) ‘ WW“! M135 43;; f _ i092 Q+fi*#3 7'? [,9 .3. . $3,”, on»: «Straw; ‘ fiflfg'gi cow 0‘ I 50 (H; 67}, £71) I “.5 ii i1 xix-4%? maid [9] Math 1E5 Final Exam 9. Page 9 of 12 7 Name; (a) Letz=1+i, (i)'Express z iri polar form. , r 7.1;11 r": [EL a: Eva-J“; (gag) :» 0353‘ j? (ii) Find zlfi. Expmss your answer in the form a + bi ' r {a ‘ g 47,7 - ~ km 1‘ L @ ? , M Ly? a e mam r: @319 (B) Find'all complfsx numbers 2 such that z3 2 1. Express your answer in the form a + bi. ! 2'3” [gfitm‘fiw ‘! a) Ffif ~ ,_ - ‘319 ' 1m. 5“ gait? 2.5;; CL at“? 22 {353%. m vi 4» t; - W375?" "'7; ‘ (0) Note that i3 = ——i. Use this fact and mm; (b) to find the oLher 2 cube roots of ~12. (n1 Aims“ mfigfi mfg“ firt‘ ( L ) a +1 Math 115 Final Exam Page 10 of 12 Name: [6] 10. (a) Show that the matrix A = [g is diagonalizable if a. 95 b. @(w: / 5""? ‘5“! /;. fi}(01"é) a sub 875%» i, Am) dzhyéwL Ciflggj 72"“ I ‘3 affl(§m&fiifiarg€£& I (b) Show_thatthcmat1‘ixA; [g i] is not’diagoualizable. ' h! h ’. F, L MW 3-- ‘ m (9‘ ‘5‘) ‘ _ 4) “a _ ' ’flwfi I15 C! {New if“ Mi 60-46 fé’rléfl’hwfiaf.‘ W; m _ “95% allay: a (égfldg (3% _ (Hm; git; fags? 51m gawégféé ' MalhllSFinalExam .- Page 11 of 12 . Name: [8} E 1. Detennine if the following statements are true or false. Provide a briefjustificafion for your answer. (a) If A is a 4 x 6 matrix, then the dimension ofNu11(A) is at least 2. (— allow- nuéfzm + five/‘41??? self; SHIch rem/€04} 557' Mm) I ‘3'”, - (b) HA is similar to I. then A = I. 7’" Swé (G) Let A be an n x n imam-ix such that A2 = 0. Then A =‘ 0. ,F M: (3;) (d) If AB 2 0 then (BA)2 = 0. T @W’ {WW (WM ‘ I Efiafi ...
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This note was uploaded on 08/05/2008 for the course MATH 115 taught by Professor Dunbar during the Fall '07 term at Waterloo.

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solutionsoldexam1 - w ' SOLUWo/ur University of Waterloo...

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