MATH115MF10P

MATH115MF10P - Name (Print): UVV Student; ID Number:...

Info iconThis preview shows pages 1–10. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 6
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 8
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 10
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Name (Print): UVV Student; ID Number: /_>g_/0,La/o » University of Waterloo Midterm Examination Math 115 (Linear Algebra for Engineering) Instructor: See table below Date: Thursday, October 21, 2010 Term: 1109 Number of exam pages: 11 (including cover page) _j 1M3 Section: See table below Time: 7:00 pm. — 9:00 p.111. Duration of exam: 2 hours Exam typo: Closed Book Additional material allowed None Circle your instructor’s name and section number Instructor Section P. Pei 001 R. André 002 R. Hamilton 003 H. Wolkowicz 004 K. Georgiou 005 Y. K. Chem 006 R. Andre 007 B. Anderson 008 Instructions CHE students GIVE students GEOE - ENVE — students ME students ME students Mechatronics students Software Eng students MGTE students 1. Write your name and ID number at the top of this page. Please Circle your instructor’s name and your section n . Answer the questions int using the backs of pug rough work. . Show all your work requ answers. The use of calculators or umber up above. he spaces provided, CS fOI' OVCl'fl0\V OI' ired to obtain your any other aids will not be allowed for the test. ma BEAMINERS’ use ONLY Question Mark 1 /3 2 /2 3 /5 4 / 5 5 / 3 G /3 7 / 5 8 /5 9 /6 10 / 3 11 /6 12 /4 Total /50 Math 115 Midterm Exam, Fall 2010. Page. ‘2 of 11 Name: [Marks] {3] 1. Reduce the following augmented matrix to REF. 2 4 ~3 6 2 4+!) 4 8 b~12 —4 —8 a + 6 From this reduced form determine the values 0,1) for which the system has: (a) a unique solution, (13) multiple solutions, (C) no solution. Justify your answers clearly. '7 . g_ ‘)/7_ l‘ l g) 4L —3 " {a A (I b q. l 9— QLHELL 3 447%. (000 all: _L/ ~3 0M”? B"”’ 01/3148; ,— (fi I W315 3% Q7 5 C) 2 b 7% (4%WVA /J a/V V4, ’ IV :4 (7% M 4L0 D/ 30- "BI/L .27)? d J b '0 C :3 l4/s/H ._.«\/L dxw (JO 0 o o —y:. limb M ' “71214644me % ff” I; [2] 2. Find deLA for the. given matrix A: 14—3—6 0447 A:O—8712 0100 044» a?! 4M; , I)” [00 :; (+3”l5?2’) [5] Math 115 Midterm Exam, Fall 2010. Page 3 of 11 3‘ Consider the following system of linear equations: $1 + 211:2 + 332+ 3133 3173 9:173 : phi-4C7? Name: H (a) Set up an augmented matrix for the system of equations. 5 Q; / / 2% ’7 Se / a. “I a 9 0 (13) Use the elementary row operations to transform the augmented matrix into reduced row echelon form. ‘ 2...; / % 124mm 0 #Z/(I 7743 C) a_ I 6/. / 0 M O / \LIQL 'I’R/ D O (c) State the complete set of solutions of the system equation, 6 (d) Express the vector 1 6’; “$3 3—0 as the product of the 3 X 3 coefficient matrix of the above system and a column—matrix (vector) 2 />3 2 O l - ‘l e L —&3 '24) [r [2' J Math 115 Midterm Exam, Fall 2010. Page 4 of 11 Name: “a 4. Consider the matrix A and the column—vector b: 1 l 3 6 *6 A_ —2 —2 —6 —12]' b“[12]' O (a) Is the vector x = _(2) a solution of the system of linear equations 0 represented by the matrix—vector equation AX = b. State Why. @ ~§a / /1 —:fan] a r; X /L] -1 F _ 757 (b) Given the following homogeneous equation express the complete set of solutions as n vector equation. $1+3§2+3$3+6$4=Q [/ /5<ol03 Parr? (If/24m ,( M W (fa/Maw 47M, X, XL x3 x " 1,: _>(L «33(3 —é'>(L/ , / 7’ 0%“ 3%,: 1an 4—ox3 ri-cvln/ 3(3): Ojlmklxi koxL/i sag : am To"; 4* ’ *v -I -3 ~C BM 0 wk 0 55C 7W ,. XL (9’ "L >L3 ’ 1" 5‘ <2 x3 — u U ( ’Xg (c) Consider the equation AX = b. Solve for x. / / 2; {a )—¢:J / l 5 Cr fill» L 3 . o o -ZV’L‘L—IL / lfllrfmp O U o ._'g AC 3 5'“ "(v —/ L O 0 3h. (9 + Db. ' "l 7 5 3 +3“! 0 :2. CD 0 7Q} U o D I Math 115 Midterm Exam, Fall 2010. Page 5 of 11 Name: a [3] 5. Consider the 2 x 2 matrix A _ 1 2 _ 2 4 Find all possible values of .T which satisfy A2 : 1AA fizz 30‘} :5 flz‘ 1/} 2 5*W4LI’L/U/ 7 721ml; E/ 7., 9g 2-6\ 3 opsz-mv/ ‘31.. LJJ 1L{ ’— ioxx 47L 5 20 [m M Q Orv/MW “9 Lu '— Zak $7» [3] 6. Suppose the lines L1 and L2 given below are known to be perpendicular. mixznlmail M Find all possible values of 17, y and ,3, 1 Ha X >4 l l _ 71 (:1 —' 4r " W x 5 FM: birds]??? Math ll5 hrliclterm Exam, Fall 2010. Page 6 of 11 Name: M 7. Lot 13 denote the 3 X 3 identity matrix. Consider the 6 x 6 matrix _ 213 313 H43 (22.) Show that A2 : 15, where [6 is the 6 x 6 identity matrix. ’2. 3:5 waif/j KQJ’S 4‘3 , r“ fl -/_W A k 'f3 “ALJ’B ’L3 [J’j/ (1)) Determine A115. / 0 1 (C) Let b = (1, 2, 3, 4, 5, 6). Determine whether the vector b can be written as a linear combination of the column-vectors of A114. 2'. 4. J Pr. % l—, M / E: 4; fl/LfDCC g: -’-> “X s S G fig (.9 Math 115 Midterm Exam, Fall 2010. Page 7 of 11 Name: a 8, Suppose T : R3 a 1R2 is a linear transformation which satisfies T(1, 0, O) = (1,4) T(O, 1, —i) : (1, —1) T(O, 1, 1) : (o, ~2) (a) Find T(0, 1, 0) and TU), 0, 1). I Spjzfic m :5 2> ll : 42:] mm : [iifl , + ‘;JO 2 :_ B 333% >397); :\ La://)_ 7(‘tu/éi % m = w (:3: m l \ ' ‘0 ‘\J H M 'a A O :3; (1)) Find Lhc 2 x 3 standard matrix A of T. [6] Math 115 Midterm Exam, Fall 2010. Page 8 of 11 Name: 9. Let A r //—5 [00 {’0‘50" 0 ( L o I '3 B D 7’ O c9 [3 0a I “NQLd—R/ 0 0' O —V’27."JW/r?3 _L S" 4 ~1fl3 V‘flL / o O / S/Qg #13, 0, O O 3 "L 00 I} O "l I / «(a 3' -l A O 3 —‘—' fl 0 d I (1)) Find the matrix C such that T i 012 0H1 2 0] :7... T h 0/1 w C4’Ai/ZD “— C :B c; : @0717 2; L: 1 _L03“/ {-1) )2“) o J I: / D 0 ‘ W (DD 7: O — 2 1 \ (J o \ ' I l 3” 0 Au] , l3] Math 115 Midterm Exam, Fall 2010 Page 9 of 11 Name: “a 10, Suppose P is a plane in R3. If the point on P closest to the origin is (1, 1, —1) give both the vector and scalar equations for P. 2 *1 11. Letu: 2 ancld: O . 1 3 (3) Find 11 - d, 11 II, and d (1;,C} C —Z,+ov‘—3 3'/ #4:” : SIG-:3 W: KO.) (b) Find 111 and 112 such that 111 is parallel to d, ug is orthogonal Lo d, and u : L11 + 112. Math 115 Midterm Exam, Fall 2010133459. 10 of 11 Name: [4] 12. Let. A = [ 1 l 1 Write A in the form 2 0 2 A = UARREF where U is 2L product of elementary matrices, and ARREF is the reduced mm echelon, form of A. q. /’\ (I/ /1/ 5 I10) Z \ , / o 3/ I k ‘2 "HQ1 4/? j: 2’ L) / /c> )“ 92,46; " 0 (o / ZXL :‘B A I E2lgl’f‘g", 6—1/Q1 E‘flL’fflL Total = 50 pts ...
View Full Document

Page1 / 10

MATH115MF10P - Name (Print): UVV Student; ID Number:...

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online