Math 152 - Linear Systems
Test #1, Version B
Spring, 2009
University Of British Columbia
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Name: _ _
ID Number:_ _ _ _ _----'-_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __
Instructions
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You should have six pages including this cover.
There are 2 parts to the test:
o Part A has 10 short questions worth 1 mark each
o Part B has 3 long questions worth 5 marks each
Use this booklet to answer questions.
Return this exam with your answers.
Please show your work. Correct intermediate steps may earn credit.
No calculators are permitted on the test.
No notes are permitted on the test.
Maximum score= 25 Marks (attempt all questions)
Maximum Time= 50 minutes.
GOOD LUCK!
Part A
Total
B1
B2
B3
Total
10
5
5
5
25
.. Math 152 Test #1 version B
Part A - Short Answer Questions, 1 mark each
AI: Determine cos e (as a fraction, possibly involving square roots) where
is the angle between the vectors (1,2) and (3,1).
(1,2)1 -: 5
e
.,2). (3j l) :SJ
~g
A2: For what value of the scalar a are the lines
(1,2,3)
~·r.o
+ 5(1, 2, 9)
(4,0, -10)
__5'_ _
-'"
+ t(2, 4, a)
'h~J
parallel?
(~S
~tt.p 1A.a~
A3: Consider the point A with coordinates (aI, a2). What are the coordinates of a point halfway between the origin and A?
(0.., /2.
.
t1 ~ ) .
~/1-)
A4: Are the vectors t l,2) and (2,4) linearly independent? Briefly justify
I.. your answer.
x
=
y
z
=
'
f
[1 1 1]; = [1 2 3J ; cross(x,y); Pick the case below that describes the object z assigned by the code:
(a) a matrix with 2 rows and 3 columns.
(b) a matrix with 3 rows and 2 columns.
(d) the statements result in an error message.
2
-
Math 152 Test #1 version B
A6: Consider a linear system of 2 equations in three unknowns. Pick the
case below that describes the system geometrically:
(a) The intersection of three lines in the plane.
€) The intersection of two planes in three dimensional space.
(c) The intersection of two lines in three dimensional space.
(d) The union of two planes in three dimensional space.
A 7: Give an example of a nonzero vector perpendicular to (1,1) .
( -I, I)
lj-. i? It ) .
M
A8: Consider the following lines of MATLAB code that modify a previously
defined 3 x 3 matrix A
for i=i:3
4·,
A(i,i)
end
Circle the case below that correctly describes the action of this code:
(a) Sets all entries on the diagonal of A equal to 4.
(b) Multiplies each row of A by 4.
(c) Sets all entries in the first column of A to 4.
(d) Sets all entries of A equal to 4.
A9: Write the formula for proja b, that is the projection of b in the direction
of a.
Q.
,b
proj! ~ - - ­
-
AID: If a is nonzero and a is orthogonal to b, what is projab? Briefly justify
...
your answer.
"
­
,'0 "::. 0
-
-
So
I
3
o
Math 152 Test #1 version B
Part B - Long Answer Questions, 5 marks each
Bl: Let A = (1,1,1), B = (2,3,4) and C = (0,1,0) be three points in three
dimensional space. Consider the triangle T whose vertices are A, B
and C.
(a) [1 mark] What is the length of the side AB?
(b) [4] Find the area of T. Note: This area is half of the area of a
parallelogram generated by any two sides of T.
a)
AB-= (2,3,'1-) -(\,1 .. 1) :: (',2,3) ,1
~"'8~ b)
At
~
(0 1,0) - (I, I J I)
~
,
1L" e =
-I
v...~
.
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-t
~
W. 3~ :
2"2
=
0, -I) ~~""f
( -II
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0
:::-
3
2.
).
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Math 152 Test #1 version B
B2: Find all solutions (if any) to the system of equations
I
2
"2.
-l
I
3
-I
-I
-I
o
Xl
-X2
+X3
2X I
+X2
+3X3
2XI
-X2
-X3
= -4
=
3
-1
-If
I
3
o
-1
o
-\
,! ~
(2)-Ul) 3
i
I( "
-~
I
'~ o
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3
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1 ::
1
1-
(- \
(3)- 2.( ,) ,
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~
't t - (4)
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~
3"
10
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~ (I J 4-,-\).
5
-
Math 152 Test #1 version B
B3: For each of the systems described below as augmented matrices that have been put into reduced row echelon form, determine whether the systems have • no solutions
• a single solution (in this case, find the solution)
• an infinite number of solutions (in this case, find a parametric
description of all solutions) .
Each part is worth 1 mark.
(a) [~ ~ / ~1]
(b)
[~ ~ 1
0
1
0
[~ ~
2
0
(c)
~]
(d)
[~
0
1
(e)
[~
0
1 2/ 0]
1 3 4 0
~1 / ~]
~h~\.e
:=. ~
j
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It\.a
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X ,~ Lt I
X-z- - -t ·
1.."1..
-=
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'1'L -= 1. + t
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Xq
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