Math 152 - Linear Systems Test #1, Version A (T/Th sections) Spring, 2010
University Of British Columbia
Name:
----------------------------------------------------
ID Number:
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Instructions
• 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
Although all questions in each part are worth the same, some may he much
more difficult - do the easy questions first!
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
l\'lath 152 Test #1 version A
Part A - Short Answer Questions, 1 mark each
For Questions A1-A3 below, let
AI: Compute 2a ­
(
~I
A2: Compute
II all
a
(1,-2,3)
b
(-1,0,1)
b
.... tt (
\0) - ( - I )
0 )
I") ::
A3: Compute a· b
-I +0 +3 -:
2
A4: Rukia and her brother Byakuya each received some money from their
parents (possibly different amounts). Then when you see them shortly
afterwards, neither of them will teU you how much money they have
received . Instead, Rukia tells you how much more money she has than
Byakuya. Then Byakuya tells you the total amount of money given
to them by the parents. Based on these statements from Rukia and
Byakuya, wou Id you be able to determine how much money each of
the two have received, and why? Circle the answer with the correct
reasoning below:
(a) Yes, because the statements represent a system of two equations
and two unknowns and therefore has a solution.
@ Yes, because the statements represent a system of equations that
may be reduced to only one solution.
(c) No, because the statements represent contradictory equations.
(d) No, because the statements represent a syst.em that contains too
many variables and not enough useful equations.
A5: Consider the augmented matrix below:
2 3 1
2 1
[
000
o
i1
Decide whether the corresponding system has no solutions, exactly one
solution or an infinite number of solutions. Justify briefly.
\V\h~tt
0--
l\UV\1W
tb ~~~C
~r~~1
Math 152 Test #1 version A
A6: Consider the following lines of MATLAB code:
A = [1 5 -3; 3 2 1J; A (2., 2) What will the output be if these lines are typed into IVIATLAB?
2­
A 7: Consider the following lines of MATLAB code that modify a previously
defined 4 x 5 matrix A
for i=1:2 ACi,:)
A(i,:)+A(4,:); end Circle the case below that correctly describes the action of this code:
(a) Exchanges the first and second rows of A.
(b) Multiplies each row of A by its row number.
(c) Adds the fourth column of A to the first and second columns of A.
@ Adds the fourth row of
A to the first and second rows of A.
A8: Are the vectors (1,2,3) and (-1,0,1) linearly independent?
Justify
~ s.
~~ Q . ~.
(~f~ duv-ec1lCf\ltS} (\O-t S~
briefly·
I~~L
J
M V
Ih'pi.9s)
A9: Find a nonzero vector orthogonal to both (1,1, 1) ~ cl (-2,3,7).
1kt- G I'oS.t p~rl~ ~ () ro'(JQ.f f7J
AIO: A lineal system with three equations and
all solutions lie on the line
j
I
k
r
t~ree u~n
-=
vns is such that
x = (2,1,0) + t(l, I, 1)
What is the reduced row echelon form of the augmented matrix of this
system?
I
(
0
-I '
-1
I
o \
o C? ott
)
( '+) - ~ ) S- .
l'vlath 152 Test # 1 version A
Part B - Long Answer Questions, 5 marks each
B 1: Consider the linear system for the unknowns x, y and .z:
+ 2y + 3z
X - Y + 2.z
x + 4y + 9z
X
9
9
25 (a) [1 mark] Write the system in an augmented matrix.
(b) [3] Do row operations (Gaussian elimination) on the augmented
matrix to change it to upper triangular form.
(c) [1] Find the solution to the problem using backward substitution
from the form above. Hint: this system has a unique solution and
all components of the solution are integers.
,
CiA) (b)
(
'­ 1
-- \ ?
t
4~'
2
f\.
-~
0
0
-I
~
1-
9
9
(
~
i
1..5
~
0
ICo
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(l')-(l)
C~)-L
I)
I\.
2>'
0
-~
- I
0
0
0
-~
lCo
l!a
\
Cc. ) """"3 ::
3
- 3'h
i, -2
2>
.J.
=-0
=)
~
: <3
"'2:- - j .
=-)
X, : 2w
4
q
C~)+~~)
Math 152 Test #1 version A
B2: The line L passes through the points (1,0, -1) and (0,1,2).
(a) [2 marks] Find a parametric equation for L.
(b) [1] Show that L is parallel to the plane P with equation
2x - Y + z
=
-2.
(c) [2] Find an equation form of the plane that is parallel to the plane
P and contains the line L .
~ ~
L:
,
L [s fur~MlM'
c'6
~ ( - I, \ ) j)
d..J,'H
~
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~
cAiv,
Y\..oY~
('2., - 1, f)
~ ::
-
0 , -( )
~ P.
-
V\
a.
(I,
ro P if
p[;.,r~ l \.d
lS
-
L'/~/-I)+-t (-I,I/~) =
~
l
(\:J)
~
(0 , " L, )
l
V\ =
-2 -1 +- ~ ::
0
-/
L-
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t-o p.
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r~ ~ P . If- tv\v~+- 0.)<;0 ~VL ~
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ovUo <JV\,hl'·.·;s ~ V
~ oLt·~G4.~SI~ ~ / oJa ~ POf WJ
(0
I ( I
~ ~~
l.)
I'S
(i"V\...
~b
r~
Math 152 Test #1 version A
B3: A line L is given in equation form as
Xl -
X2
+ X3 =
+ 5 X2 -
2XI
X3
=
2
( l)
1
(
1. )
(a) [1 mark] Is the point (1 , 1, 1) on the line? Justify briefly.
(b) [2] Find a line direction vector for L .
(b) [2] ·Write a linear system for ( X l , X2 , X3) whose unique solution
is the point on L tha t is closest to the point (2,1,2). It is not
necessary to solve the system you write down .
(~ )
~
"f.. =-
C"
l"
I)
) "\ - XL
~ hVSt- e.~ dvt ~ is
Q...4Q..
So \t- {S (AJ>+ mA- k
q
-
- ~
1-
n 1-
/\
A... ~
~
1W po ,vJ i~
((,')
( ()
2,
('1.-)
cv{ru~,
1 - (L,',2)
Q,
~
:)
-
1-
S
I
\
-
0
\
VV\ \J~ ~
1
cLv~sl-
tu
VV\V\t
L:£ - l2, IlL))-=-
/\
A.
I
0
SQ
S~ sr. --esJ. /
t-
rI\.P
~
T~
~ 3 = '.1
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