midtermsolnsmath115fall07

midtermsolnsmath115fall07 - Name (Print): UW Student ID...

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Unformatted text preview: Name (Print): UW Student ID Number: UniverSity of Waterloo ‘ . Midterm Examination ‘ ' f" Math 115 (Linear Algebra-for Engineering) Instructor: See table below Section: See table below Date: October 16, 2007 Time: 7:00 p.111. - 9:00 p.111. Term: 1079 I Duration of exam: 2 hours Number of exam pages: 8 Exam type: Closed Book {including cover page) Additional material allowed None Circle your instructor’s name and section number Instructor Section F. Dunbar 001 CHE students R. Andre ' 002 GEOE - ENVE students M. Kamensky 003 GIVE students "I. Moffatt 004 COMPE students F—S Leung 005 ELE' students S. Wu ' 006 ELE students J. An _ 007 ME students I. Mofiatt 008 ME students F. Dunbar 009 Mechatronics students R. Malinowski 010 Software Eng students R. Andre 011 MGTE students Instructions Question 1 /3 1. Write your name 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 are allowed. l3] [5] Math 115 Midterm Exam, Fall 2G0? Page 2 0f 8 Name: 1. For what values of a does the homogenéous system ll o 12:6—ay cmi3y : 0 have mere than one solution? P &2M4 %+m4xOCfiflkj:::;r;:§;7”. QiqlkC/vL);t o ( W; A— Lg Haj ijwe :5“ ‘ \ 2. Find the general solution to the following system of equations. Write your ' answer as a vector equation. ' w+9y-52 = l] m'l'2y*2,:0 3wmy+2 = 0 r“ .. Jr 3P: C? *5”l I if 4r k 1 ’\ (‘- O r é , l Jae-r” o ‘13 ‘ (S "-I 5d "9— <1“. ’I‘f‘Ql l _UR2’T'{8‘3 , \v L)‘ f l o\ ' -5 lo cl vi 9" U Jr if C, o i; "am 0 O O 1.,fifiLlfi‘ - ‘v Vi >L: “WE; \/ (l l l 0 3L ‘1 r W}: [O l “HI/$1- : ') J." 3 0 C9 0 J d— a” ." "‘ , u 3 W \ mt; is"; i ’34?) ‘f’ S? I __AL“/‘; "I" Math 115 Midterm Exam, Fail 2007 Page 3 of 8 Name: [8] _ 3. Let P = (3,2,k), A = (1,4,1) and. B = (1,4,7). (a) Determine the midpoint M of the line segment joining the points A and B. A+BE,L[m—KU+(wfi¥fl M 1 L C 5 tr . r I _ f l (gm/’10} A I J J L (b) Find the projection of vector (—375 onto the vector E, projIH-afi ALB : 86—4 5 (£21.?) I (5‘33) :1' 0/ Li’- / C» ) O 33> 5. ( 3 2/ A) Wm_—m~wwm'“’wmMM:Hg . C9 ye 73’“) M: (w x #5 //‘{:’Olygéé)nj/L l I (fflfl '2. L 'L 2.. 07% H; 5 2 4+3KKQZ5 /3 4) (C) Determine the value of k so that this projection is equal to the vector AM, 7 where M is the midpoint of the line segment joining the points A and B. a? :: flaw—,4 : (C '3’: H!) #- (/ffing) :'(O/a,3) geéé (946): MM [ax/,é/ V ) Math 115 Midterm Exam, Fall 2007 Page 4 of 8 Name: [6] 4. Let B be the matrix (:1) Find the inverse of the matrix I i 0*L' /'w .-_._{ '"f o I Q / A I .. its»: [0‘ ('90 0 L“ , ,.r=m“":‘[‘ 7;” 0‘93 -1 —J' O ‘J‘r J r a mold} \QUL' laud f g vamp [pr 90’? :0 \ new ’ )‘f'LLJ J o o -i 4.314, 0*! \\lo’;:lflgumwfi \1 —-| "‘ IR3+R1 C13 94* 0--E 5° .‘L- 1 a a“! n-«2 Exam: A \ EL 4 \:'£L’ 7/- K I‘ .— ‘ ‘ \r IR?) . ‘ w \ \ e7 ‘5 ‘1 kt 0 \ o \"\ ‘ \ :9 o \ x .ch ‘V" (13) Find the matrix A such that 1440441) 2 B. Math 115 Midterm Exam, 1933112007 Page 5 of 3 NsLme: [4] '5‘ Let A?“ H- Show that A‘1 exists and then write A—1 as a product of elementary matrices. ' we r ~ i? 'r a “- 2— t " RE": a} go I} “a? )5 4' 4 El} .4. .. .—I ~. F u l r“ L9 2, Emwmkv S I w) “EMA- Lim I #35:)? [a .] “L f riffs?“ IL, .., 1“ PM J pm, 23,9 4 '3‘ "L ‘ ‘ J— 13 Ewan; 4 c:— 4-— fiflym’ T59 I. .__ ‘ //i‘ ’_I at [I ’0 DJ \_“_£M-:f/ ,5; :L6 t] 0:1 1; Ir: 1 1 0 41 E3] '6.1fA= 2 i g _1 calculate det(A). L U 2 —1 0 EIfFM/IJ @530?th 1mm” . '5‘ I ~.\ .— 1 a I Adm- a. #3“) c 1 [5}! I“) c; r! L) 5 ,3, (F! +13 5 O [3] 7. Let B be a. 2 X 2 matrix. Consider the linear transformation T : R2 a R2 given by T(X) = BX. Suppose that we know that T<m>=m Tamm- "Tum mm no,» w) r TCIIE,) zr fear-CF?!) Find (3,!) + 6’0,” J ‘n g: {'51. a} / Math 115 Midterm Exam, Fall 2007 Page 6 of 8 Name: 2:: [10] 8. Let T be the linear transformation T = m+-y and let S be the y 23,! -— :c transformation from R2 to R2 given by reflection in the line 1; = —;r:. (5.) Find the standard matrix for the linear transformation T. f5 / (’10):- (2/!)"!> gtfg/Hehadc} E'Vhfl”?lj”ll)" 770,12 scuba”- its] #! -2} la... (b) Dzaw ‘3 picture which indicates how 8' acts on a point (3;, y) in the plane. -Wfl....._g _.__?.§r- \ 'fi‘x \ K5 L3_:r":\n (c) Find the standard matrix for the linear transformation S. SKIID)‘ (014) SCaFE}; (mg-IQ; (d) State which of the compositions S o T and T o S are possible M 2 on {m wow w\ 0 2-11 L... 1,..1354’1 _ *‘3. D —i . ‘ “175: E. 7*. LJ‘ a] 42:91:: me: (e) Find the standard matrix for each of the possible compositions found in (d)- ‘ d— 1 > [it Tagcl,0) C “id—(0’4): (Div-if Togfcli) : T (Whig); (_-L_/ t;) E) l Lu 1, I [8] Math 115 Midterm Exam, Fall 2007 Page 7 of 8 Name: 9. A square matrix A is said to be idempotent if A2 : A. . ' 1 U . (a) Is the math): A : [ ] idempotent? Why or why not? [El :3 no] ; It 6 fiamrfligurf i o r l (b) If A = [1 0] , for what values of a and b is the matrix A idempotent. a b mew _ Eln‘?1;n§' (2cm, J Lab?“— L‘fi'ttl’ ‘3 J t x ‘_ [3 IS 0 . w k 2",“, (I’D-H.) _ wth H b” :- LI—tw m ecu—We - g: o :5 a, w“ {naclrwrt-«u? DO] I_ ' Wb:\ 13:25“; afi~_mlk “ l O (max: Maw Etotl ,1 . are. (all,\ .: (3 l\\ (c) Show that if A aficl B are idempotent matrices, and it AB = BA, then AB is also an idempotent matrix. - “Le x 23 Ware 3 ABC Rae" ~"""> A -f ‘r 3 ) {flmmm : (ABHEA) : Ft (833) PF _ i T j_ 2 t r.“ we -"- «MM 1 A 8 Pr (6, HA/ 4‘ 5 IS? A 3 , _’ ‘l /A {W A ; C ft B ‘ gm ’3‘. (VJ/bum vaifim 3 Math 115 Midterm Exam, Fall 2007 Page 8 of 8 Name: This page is [or roughwork ...
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