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MATH115MF09S

# MATH115MF09S - Last Name(Print)“_First Name(Priﬂ UW...

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Unformatted text preview: Last Name (Print)“___First Name (Priﬂ UW Student ID Number: 5 University of Waterloo Midterm Examination Math 115 (Linear Algebra for Engineering) Instructor: See table below . Section: See table below Date: Monday October 19, 2009 Time: 7:00 pm. - 9:00 pm. Term: 1099 Duration of exam: 2 hours .Number of exam pages: 9 Exam type: Closed Book (including cover page) Additional material allowed None Circle your instructor’s name and section number Instructor Section W 001 CHE students m QQg ENVE — GEOE - CIVE students " RM, ”00.333 GEOE — ENVE _ GIVE students akin—Ig- 004 ME students m 005 ME students m 006 Mechatronics students w 007 Software Eng students # 008 MGTE students FOR EXAMINERS’ USE ONLY Instructions 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. The use of calculators or any other aids will not be allowed for the test. Math 115 Midterm Exam, Fall 2009 Page 2 of 9 Name: ———————————__—.__________—————_‘——— [3] 1. For which value(s) of the constant a is the following system of linear equations consistent? H :: x—y —ax+(a—1)y—2z y+22 2a II [I H \ “l '0 i Q75 ' ' , o , , O ‘ \ l 0v I I *Q MW“? i 9Q wt Q‘QK <13 ~\ '2 [09+wa V} x o l 2 it 02*.10’HSFO (orncm-éi‘m dirt—l 9! \ 6VS’r-ﬂm is wr’iﬁifﬁnrw l va SqSicnn is <yoneasrcrﬁ“ ’%v qu VGer a.eaz,f Voir’lCVB Q #2 " i 1/”! {\$3 [4] 2. Consider the system of linear equations :c—2z+w = a y+32—w = b (a) If a = 0 and b = 1 ﬁnd the complete solution to this system expressing your solution as a linear combination of vectors (if a solution exists and is not unique). ' l 0 +2 l C) XﬁZt—S o ‘ 5'4 I v\$=\»3b+s <2ze NCS 21 ‘3’ '2. """ *l ‘5 + "’3 45+ 1 S x" l _ ' :g "7 g l O / fZ/ . (b) If a = 0 and b = 0 ﬁnd the complete solution to this system expressing your solution as a linear combination of vectors (if the solution is not unique). a o~o.| 0] 1‘ Zi’s =~5e+s 01264 0 Qgt was 2 «s s .94 Ci \ l J 0 I Q | O”l“‘2 @1495}; [254320. l Math 115 Midterm Exam, Fall 2009 Page 3 of 9 Name: \ _,—- 0 l [8] 3.LetL1bethelineL1={x:x=[O]+t|:—1:l}inR3. 1 2 (a) Find the equation of the plane which is perpendicular to L1 and contains the point (1, 3, —1). T j6¥ hm 33:7? Cl: Flaw; ’= [-l "l 21 snow ~eunl1'ir9m o? a Wane; \ . L1 ,, at 322;}; + at \$313+ c. ("e-wen '20 t / 7%,? < a ”0‘0 ”:7? , i - . WWW \ (1” on \ (wan wane o lmwm 2%: , .n — k5+ 3) + 24¢ l 3 O ; .egfuwsow #91an 2 1— «3+ 2:: =7 , 3?] (b) Find the shortest distance from the point (1,0,1) to the plane found in part (a). ﬂute”) l (523:3 ‘13wa re) a wmmm “W 4W? Home) 1* .2 :i “7‘ ‘ L7 d was my“) Y4”? ESE \‘Z‘a Cl ‘\/“6 (BED? .L \-/\ 55'- 8“ :1, E \ \ O3 ‘ ﬂab-AR). _,:_ L} ' 7, 9, " . ,‘ The 5mm“??? 5 “N l + 5 ,2 w. dra‘mwa l5; ' W} : 7%[1 ; O] 2‘” [3: -— *5 ml liq" ~ 5... -wob- ”“2“," """" , w; ... 2:31 W. ‘7... :‘~ :- 43. ”4% J3 mﬁ : PPowPQ [0“3’) «Cl‘f 2 3:2 “:1 ["2 7— 93' (c) Find the point where L1 intersects the plane found in part (a). / gt 3: alt‘tﬁ4} (My 2 L P \anz: ) 4" ﬁx”) ’ * .2 ' cf ..... :2 / sobi‘mtwvc been. into ﬁrst mum-low \ l , ll 0 "l" 5:5, <2 4 —5. 3 ’5? a; 5i (9 J. (a Hat POM/W :2. (via, % "2:35) Math 115 Midterm Exam, Fall 2009 Page 4 of 9 Name: [3] 4, Three matrices, A, B, and C are given by 12 3 0 5 1 2 6 ' ’ km em ermiwm Compute ACT + BT. ET [9; 2, ACTiiéT -; [£1262 5: 2C + 0 L) .3 28 4+21 o \d g o 5 1 6oxu+5x. + 6 2 is Mezxeg \/// ‘55 , \Bf’SX; [1] 5. Express the system of linear equations 3:1 — x2 = —2 2x2 + 3563 = 17 4331 + 2x3 2 14 in the form of a matrix equation. iii (a) Find the rank of A. ' 7 // P G) QOW‘ QQdLAC‘CM w“) \ “l 0 ~ 0 .L o r -L ‘ ‘ .L 10‘2” 1 o 2 ngre are 2 1 X 9,, C) \ .L " m '7 — Q 7” O \ Z . .1. ,1”? 35515.} “the row/UL? 3 3 (b) Find the (2, 1)” entry of A2 + AT. M3+R15 Wrso‘finia' g 03 i3 ~33 '2 :3 :3 3; o 3 :2 2' ( '3 _.| ‘ O '1‘ 3 I 4 A“ “i 2f?) Ci “IS 1 o 1 '2 : ”2 «3 2 2 5 I // Ci “H 2’) ' V Z The C 2.401“ «(it"Vsi'i’q mm} ’3: i ‘3 2 (3oz Li W i”: “i Math 115 Midterm Exam, Fall 2009 Page 5 of 9 Name: [2] 7. In the block matrix C = l: g 103 J, 0 denotes a square zero matrix and I an identity matrix. If B is a matrix such that B‘1 : B, show that 0—1 = C. ,‘ «:2 “9‘ ﬂ I ' ‘ It?) 8, C, -—;C\.dlczi g0 - .1 ”clertc KE‘O 0 1 mm C a“ x O 2 I O y . 0 15 D W‘ i :11?) D 2 37:?! [12> o l m \/X ”W Raﬁ. 5t) 3‘ O I: {l ‘ r [I 0".] sqbshhﬁl ﬁg \/ r O 6 ES" 1534 «’2, ‘ i \3 C_l :: 'X, C) 1 .3 6:: \$31.?) ”) \3 b “)0 \+ O x?) I) V \5 . \f 1 —1 3 [4] 8. Let A be an invertible matrix where A‘1 = 2 0 5 —1 1 0 1, (a) Solve the system of linear equations AX = —1 where X is a column 3 1 matrix (equivalently, Ax = [— 1:! , Where x is a column-vector). 3 w < M 3 Eli Will é [9—, "oi Elli] ; H a / 7/ '7” 1 ~1 2 (b) Find a matrix B satisfying the matrix equation AB = O 1 . 1 0 1\(’AB* l “1’1“ \ K o \ i \ 300 Yéwi’rdwwli: 03‘; iLlDQ 6* "‘5 1~\L 10‘; o. H —\ \0 100 Bf a»; f Math 115 Midterm Exam, Fall 2009 Page 6 0f 9 Name: [3] 9. If A and B are n x n matrices such that AB is an invertible matrix show that B must be an invertible matrix. H AB is; cm mvmwte matrix; vie-i (MB) \$63 d<H A a—m 8+0 Which \mph «es thaw C‘f‘r Pvt 0 ‘ and clerk E3 "*0 , 6006 B is; {a sqmam math/ix, C (Naif) , GYM dei- Emg B \S‘ cm mv—ev-Hbse mmmx \E\J 1-—1 l [3] 10. IfA= [:2 *1 2] WhereC7é0ﬁnd the inverse ofAin terms of c. 0 2 (2 LA 31’) [1: Nil l'ii \OO i~ii \D:D 242 O\Q*ZR\O(ov'2\O 02C 00\x:0\9§00:v21 ‘VH 1004910. \el—Iio 0 i0 m”) \0 S“ ‘” 1 ea 0 »1.\ O C) D _, .37 x/ _2. J 7» 2 ‘2; c O t)! % E‘ c vi, ”3 so i A a ”’2, i (3 \‘x :7 L) “AWL Math 115 Midterm Exam, Fall 2009 Page 7 of 9 Name: l 1 0 [5] 11. (a) Express the invertible matrix A = l7 0 1 0 :l as the product of elemen— 5 5 1 tary matrices. m m p m 09 . ' A ~ [ l A 3 (523551)” ' “3 A 3 E r ‘ E2," (1)) Express A"1 as the product of elementary matrices. #5515235" k 2:1") A": 1~l 0 \0b 1/ 77 010 - L ”M"? . U10 ' " «31 12. Determine Whether the transformation T : R3 —-> R2 deﬁned by is : i i = r i is linear. Justify your answer carefully. ‘ 11 TOO ‘5 0‘ “Vii”: Us" “Weds? \$3131“ MOM ﬁlm: it pm; 35w {'7 is: '5me WWW P“ CWWV“ mad midst am; Waning; ~ («0 ”rm: % 3) :- Twc) was)“; m T (ax) = (11063 T ﬁlial) \6 induces 10% aw, W‘s-6‘31"?“ {0 “l 0)] um - an” ~ i ““ w as ”M * """" "(la “ll lax“ 9 *9 "M “ill 3: HtﬁﬂlllW‘ 35“" l . 1 rt 3,32 71% ”Va-2J1 T C 1'4 Lj‘) :3 T {"1 3+ T {‘95) l/ TV] 1 6 AW (11M 9&3» a”? (”i e131!“ 4“" 35% 53' We” 3: 0mm m (Conﬁnufd 0M?“ ‘ Math 115 Midterm Exam, Fall 2009 Page 8 of 9 Name: [6] 13 Let S: R2 —-> R2 be a counterclockwise rotation through an angle of{— and T: R2 —> R2 be a reflection in the line y— — :r. (Both of these transformations can be assumed to be linear.) (a) Find both the matrix induced by S and the matrix induced by T. surge-9m" 1:500: M {3):} [1160 TCeg) jE fl] math“ ‘ﬂd‘MCﬁd EU 3 ‘ \ 1v I i "l r ‘j A:/[ ] : lliﬂi ’ ﬁnd ((2; ‘ ' >< awe->5 vvvvv ﬂ ~ ‘ mm min a?” ' .i - iv ' ’ [of Jfﬂ 1’ +3 aims} 8- 111 3:31, ’ 7 (b) Find the matrix induced by the transformation S o T o S and use it to evaluate S o T o S ([31] >. . SC“: (sum) :1 1:71 g 0 T ° Slim) 11 Am at Merrill fill? 2;] " ‘lli? l Hid ”idlﬂ wwmvmjnaucﬂ3bq :[Zélla Cf 3%l Total. = T, [2L3 50 pts 41% soT°§l31l3FZ%ll21 .: [:11] " QaAGﬁ \$1T1i Math 115 Midterm Exam, Fall 2009 Page 9 of 9 Name: THIS PAGE IS FOR ROUGHWORK ...
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MATH115MF09S - Last Name(Print)“_First Name(Priﬂ UW...

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