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Unformatted text preview: Name (Print): W
UW Student ID Number: University of Waterloo
Midterm Examination Math 115 (Linear Algebra for Engineering) Instructor: See table below
Date: October 14, 2008
Term: 1089 Number of exam pages: 9
(including cover page) Section: See table below
Time: 7:00 p.m. — 9:00 p.m.
Duration of exam: 2 hours Exam type: Closed Book Additional material allowed None Circle your instructor’s name and section number Instructor Section
Z. Y. Sham 001 S. Wu 002
CD. Roberts 003 R. Andre H y .. E. Martin 006
RY. Roh 007
S. Wu 008
M. Ghandehari 009
R. Malinowski 010
R. Malinowski 011
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. CHE students
GEOE — ENVE students .CIVE students students students) ELE students ME students ME students
Mechatronics students Software Eng students
MGTE students. Question Mark
1"”? [2] Math 115 Midterm Exam, Fall 2008 Page 2 of 9 Name: 1. Find the (4, 2)th entry of the matrix C. _2 ‘
0 1 g 2 1 _ f f WA [0 _1 _1] o e, . _1W) r7 i " 5/ (“> "'45
\.,y/’ “ [4] i2] /,.,/' ' ‘2‘
my} T i h6/Wﬂj
i; E r i
G ‘5 i v. 2. Let v = (4,2\/§,—5). (a) Find the unit vector 11 which is in the same direction as the vector v. 197 mu“ where u is the unit vector found in part (a). (b) Compute — /" /., M? l ‘" 47:??? M?
(337140, «.9
m 3. Compute the midpoint between P(4, Zﬁ, —5) and Q(1, —4\/§, 3).  NEW
(IL/1
" : (Li m e§+ (“a war; ’3‘) 2’. M" (“3“ mi? ~ij Math 115 Midterm Exam, Fall 2008 Page 3 of 9 Name: [4] 4. Solve the matrix equation A — BX : C for X when 1 0 —1 2 —1
0 1 0 ,andC': 0 1 .
0 —1 ~1 1 2 1 2
I?) x“ A: —1—1 ,B: 0 2 J K'\ l5] [3] Math 115 Midterm Exam, Fall 2008 Page 4 of 9 Name: 5. Solve the following system of linear equations. If the solution is not unique
express the complete solution as a vector equation. —y+z = 1
2x+y+2z = 0 6. The vector v = (1, b, 1) forms an angle of g with the vector w = (1,0, 1). Find
all possible values of b. ' Math 115 Midterm Exam, Fall 2008 Page 5 of 9 Name: [6] 7. Let P(1, —1, 1) be a point in R3 and L be the line a +td where a = (2,1,—2)
and d = (2, 0,—1). .A <7 A(a) Find the poi on L which is the closest to P. WWBWWW //
(F ( l ’1 (’1 + 42+
, “WNW” W 7'
\ 7
w: 3 7% 904411615? (Jig'1‘?me which 3eparm€~z 41%: W?" Math 115 Midterm Exam, Fall 2008 Page 6 of 9 Name: [6] 8. Let P(1, 1,1) and Q(6, 3, 4) be two points in R3. (a) Give a vector equation representing the line passing through P and Q. [6} 9. Give the matrix of the transformation T in each of the following cases: m/ //
(a) T : R2 —> R2 is rotation through 7r/ 6. iii:
__ , a a re ~ r— » l i 2 CD 3 7% "” 33" ﬂ 7/4 2 ii “in
I .L
2 Math 115 Midterm Exam, Fall 2008 Page 7 of 9 Name: [6] 10. Find the value of the following determinants. ,7 \\ 4 1 :1 %’€j\ “é_5‘§~<%21¥<‘§\l3
(a) det 2 2 3 0 0 0 V f 9. "5 4+
, \ '2” ‘ r 5 /
1 l \ a b c
(b) det (1+1 b+l c+1
a—l b—l c~1 :3 E C
\ I l WW 5"?” 3‘ {rim in i ’1 '4 ’1 MM. I  ‘ (v k“) e, ffo Math 115 Midterm Exam, Fall 2008 Page 8 of 9 Name: [3] 11. Express the matrix A as a product of elementary matrices multiplied by an
REEF matrix. 12. Find the scalar equation of the plane through P(2, —1, 5) that is parallel to the
plane 33: — 72 = 5. [3] Math 115 Midterm Exam, Fall 2008 Page 9 of 9 Name: THIS PAGE IS FOR ROUGHWORK ...
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 Spring '07
 DUNBAR
 Math, Linear Algebra, Algebra, Vector Space

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