2003 Exam 2 Fall (2)

2003 Exam 2 Fall (2) - Page l of I agree to abide by the...

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Unformatted text preview: Page l of? I agree to abide by the provisions of the Univ this exam. -..a \J Date: H— i *2 - “I INSTRUCTIONS: (1) Read and sign the Academic Honesty statement abov (2) Including this cover page. this exam has 7 pages. Veri problem. Do NOT look at the questions! tier one page(two sides) of notes which must be handed (3) This exam is closed book in with your exam. Be sure your name is (4)Communication devices of any kind (eg. cell phone. pager. etc.) may no No such devices should be visible on your desk. them off or give them to the room monitor. (5)The use of a calculator is prohibited! All steps must be shown (6)You will have 90 minutes fer this exam. warning has been given, put your crib sheet inside the (7)When you finish the exam. unless the two-minute exam. hand in your exam. and ieave the room. (8)0nce the two—minute warning is given. stay seated until time is called. at which point you must stop writing immediately and stand up. Continuing to work after time is called will result in substantial penalty. e. Unsigned exams will not be graded. fy this now and raise your hand if you have a Problemi. ' r15 pts Prob4. _ f20pts Prob 6. ' ilspts f20pts Problem 2 ll Spts Prob 5. Problem 3 t7 /15pts TOTAL SCORE - . HOOpts possible GRADE % Page 2 of7 "Problem 1(15 pnts total) EC L A 1-- 1 I x I \ I 3 '1' A. IfA= —1 2 and B=[2 ],findB2AT ' 2 3 y r 1 4 -‘1 "- U r-‘l ' :I ill. 43 BI an r r- t. t t “H v H EL it L s; «1"; L an to m‘ ~r - ,f_ —- 1'“ 1 '2 1",; " i‘- H E' t 1 1mm ‘ u “t 7!“ Ell, brawl“? B. Find a unit vector (vector of length 1) that is orthogonal to both at : [-1 ,2,2] and v = [-2,1,3] “flu ' ' ,, n - . mu: ‘3 ' ' :2; \g -_—L_\-\\ _-1\- —A . . . 1‘ _- . ‘11. \P-Ixi "". vii 'H'Jfih ; ‘1 U. -— 'l x l . '3 _. ,3 l . . f--.._._ .r.. so? ._ C. What are the parametric equations for the line passing through the points (4,2,1) and (1 L1 ,0) ‘2 m J" J \J . 1’ 'L . . r. -flr'qlii-A “;\[.Ap. —-. -.‘T 1 {‘1- { r_\J l” \‘.\I. l\‘\ {331‘ \‘ II ‘d Page 3 of 7 Problem 2. (15 puts total) Select the best answer to each of the following and circle your choice .2, \ . ": - " ._ .A ~! 7-. 1 - :L I A. Which statement best describes the matrix 0 l A 2 —1 h») h -‘ - . . . 1 0 l 2 L- ' E‘I.._atn?__b_.l-,..‘ 1. rows dependent, columns independent I I‘ m f 2. rows dependent, columns dependent I rows independent, columns independent ‘1 " Hows independent, columns dependent B. Which statement best describes the following set of vectors: l , 1 , — 2 1. linearly independent -v_. '- j - g — i} F @linearly dependent i e " C“ " ‘ i”, 3. a basis ‘ 4. possibly the column vectors of a system AX:B which has a unique solution l 'Q ._ 5. more than one of the above is true R I L _ F1 0 2 01 C. If the reduced row echelon form of A is 0 1 — 3 OJ then a basis for the solutions to AX=[0] 0 0 0 1 ” ' - .. . i "'I c _ I. , .,__, would be: 1 0 0 ' A 2 0 0 ® 1 4. There is no basis for the solution set DH‘UJ D. HA is 5x5 and det(A) = 2, then __ ; .. c. J t " ‘- _ #14 det(AT) = 1/2 it" ‘ a; ié-detoA) = 486 .2 34am") = 2 '@det(—A) = 2 f E. The matrix equation AX =1 where A is 4x5 will have a solution for X if rank of A is 4 '2’ . '“ 2. rank of A 18 5 3. XA : I is also solvable 4. cannot have a solution since A is not square Page 4 of? Problem 3.(15pnts total)U5e Gaussian elimination on the augmented matrix to SOIVe the system below. If there is no solution, indicate clearly why, if there are an infinite number of solutions, express the general form as a linear combination of column vectors. 2X1+4X2+X3+7K4+X5=5 X1+2X2 +3X4—X522 X1+2X2+2X3+11X4+5X5=4 1 L“ iP‘u 7— D “I; i _ .L 11 5-1151 ' ' | _1 L V‘K—fit‘. rt ',_ O 13 I‘ l | C?! D l I 5 ‘l 1*.- r- t: l (3 ID I e; '(1 'Ful:.b , KI {it ‘-_—""I-. xngt'fiw P: '|' igrg 43%;,21 11‘— f-- _. M 41L¢ ll’i'd—W’H” ,t AH_KI Jlo' “l - 2 "2’4 _ 1'15"; 415‘ 4:L1-\l¢'t~’¢—\¢'L .- 1/ F ’ 7" 3 j 0‘ ‘ '. I I *Kt -3 -1 1&2 ' l E D c- ll I, I',‘ '3 l J \ir\- ‘_ rx '-. _1_-. Dollbc’ \Do\oo ® a ‘ ‘ 1 3 o 1‘, :3 9—1 .‘ 1,1», 1"2iu0'l DIE-IDI 3.421 ‘VZFELF‘ F r'\ I," ‘\ \. ‘ l D o 1' D ID _‘ I: I C I: ; =3 ‘7 gtoxuo‘lo‘paloiooiJ-LD *3 0 L1 9 o \ II in 0 Il ‘3 :3 HL‘ -.: L '3 "- ‘1 :3 '. 1 a .J P) 1. 0 0—1 \ '3 ' .v P‘ \ vi \ l '1 \ atflk ox M o Q‘s-w v a VI w -— .. a «raw ! U '; -'51~1"f‘- mil ', L} 0 "-. j“- ,‘ r \ D u m o [-<'7 or; Eh; ‘ ~,-'1rsrézr"'1.r<‘ lfl .r' i _\ _ Y;‘# {‘1 D '7 ." :- (hf/C) ' ‘Hv 3!? Jig [ 0 qt; 0 7’311 «H1 "Ll? o I; IN \ “ 9"r ° 0 :67” “'FH' H10 M "5!"; H1 1”?” 1 Problem 4.(20pnt3) Let A = 1 2 a) Find A“ and B4. 0 O 2 I andB= 1 3 b) Solve the following equation for X: (A + B)@_1 — I. \ D L.) i 0 1 I a mi L '_ F—Nfi-‘s J/ '| a/ f -/’ 1 0 0 DMD :1)X=BA_'X + [(A - \ o u v. -1 u x ,\ . x. w 433C: r _'\ 2 a ’5 ’L 3/ 1 cc: ~: - x. ’6’ v. a I) up '10 -_c. r'\ o 1 O :‘> 0‘ 3-1 =7 l) u 1 '32: R2 fr. 0 917-0 . 3-1 7.; "'re'c .5 ‘v 4 _ ‘io o } ll‘3,l “5— 9-1 wfiu“ -—"'L 4 5'}; r _ _ 4 1 < ._ .3 U \;r r i — 4;». I . 9 E. tin-L s' “I I 3 l —a |. .r rg‘J—Q‘EII/ 135‘ \ ' \«E' fl gi‘oaim Pagefiof'] 1 _ '53“ 6 0 2 s o " aw Problem 5(20 pnts).LetA= 0 2 0. == 1 (11:3: 0 —4 0 0 "W D 3) Find the eigenvalues and corresponding eigenvectors for A. b) Find a diagonal matrix D and a nonsingular matrix P such that A = PDP_1 . Note, you do not need to find the matrix P—l. a : E’TDEV‘ #111 o -'l l 0 0 f-H-z. o -\ "‘1' Z c: \ 0 "3 "L D o ‘ u Page7of7 K! -x 1’5 ‘ .—._ '— ‘— Rf. .1. ' (H 'k 2 O 6 r: u 1: E?1 (1. Problem6fl$pntstotal)LetA= 0 2 0 . 2,; r U f _- m a M in? ' _o ‘2 '31 6 0 2 i fix?) ‘cr \_ 1 3) Use four iterations of the pOWer method beginning at (1,1,1) to findanestimate of the largest Z r :1 ,, eigenvalue of A. o + 3211-. b) Use four iterations of the inverse power method beginning at (LL—1) to find an estimate of the smallest eigenvalue of A. L o G " 1 8 _ _ _-.' ’ -‘ ® U..:F*\|ea: a 1— b ‘1:[11 ‘11-i1{L1J 6 o 'L 'l 9 J J ,1 .. w “1 1 ‘8 — p r l . I Liz—.Pfiu'. LY "L :5 “Ht 2‘: 1/1]} 51—": ‘11 *lli’lél // i g o a! s is; ‘ J 1_ a C, 'L '1“ I n g ‘ - \n r l' u; FOO-.1. * 1 s iilé‘] "ff “J > 53'? 1 i HIGH] 1‘ \J o "L J E t, hbg'fir‘ ifirgn ,i 7.1 1“ uw‘; R93: 0 .L ,3! Meg ' '. W31 ‘1 6L1 3 “UL-.1 as] re 5, '1! i E e {_ l T __fim\ 2 z 8 ‘~ 9 [z o 1* r T1 0 61:”:3: rw ' «a. .151 -' — R ’ n -. t) l ’n n 1'. ‘I o “z. e: -v 3‘ \ _ h i i: J o F; ’ I ‘1‘; [I bniglyzbfij 1 :- uhisl :3. J L :33] ,1 1 57 G L “ 1 “ / u. _» F; “Van /’ L ".73 . ;/ i! 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This note was uploaded on 04/15/2011 for the course ECE 2331 taught by Professor Barr during the Spring '08 term at University of Houston.

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2003 Exam 2 Fall (2) - Page l of I agree to abide by the...

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