Math 152 - Linear Systems
Test #1, Version A
Spring, 2009
University Of British Colum bia
Name:
-----------------------------------------------------
ID Number: -------------------------------------------------
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
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
Bl
B2
B3
Total
10
5
5
5
25
:v1ath 152 Test #1 version A
Part A - Short Answer Questions, 1 mark each Al: Are (1,3,1) and (0, -1,2) orthogonal? Briefly justify your answer. ( l )1,1).
(a 1-\ 1'2- ') = -. '3
-t
l"-Ot- orhA..oj ~
2 ::: - I 1= 0
A2: For what value of the scalar s are the vectors (1,2,3) and (-2 , -4, s)
collillear?
A3: COllsider two points (ar, a2) and (br, b2) in the plane. Find an expression
for t.he midpoint of the line segment joining these two points.
(
~ ("-. t- b. ) , ~ ( Ql- + b~) ) _
A4: Are the vectors (1 , 2) and (3 , 1) linearly independent? Briefly justify
your answer.
cU- ~ cA..ivt__c.l\t.ivts I V\e
t{ ~ O~,
olJlk (~~ ) :::
-~ +0
A5: Consider the following line of MATLAB code:
A
OIL
~~
(~V\ 'd 1; ~tt
~~)/~'t
~ /..,,~.
= [1 2 3; 4 5 6J;
Pick the cnse below that describes the object. A &'isigncd by the code:
~ a mat.rix with 2 rows and 3 columns.
(b)
11
mat.rix wi th 3 rows and 2 columns.
(c) a row vector with 6 components.
(d) the
s tatem(~nt
results in an error rneSS8.gc.
A6: It is possible for a liw ~ar syst.em of 2 equations in 3 unknowns to have
(circle the correct answer below):
(a) only a single solution.
(b) a single solution or no solutions. ®
no solutions or an infinit.e number of solutions. (d) a single solution or an infiuite number of sOlUt.jOIlS.
2
Math 152 Test #1 version A
A 7: Give an example. of a 2 x 2 matrix with no zero entries that has a zero
determinant.
J
AS: Consider the following liues of MATLAB code t.llat modify a previously
defined 3 x 9 matrix A
for i=1:3 A(i, :) end Circle the case below that correctly describes the action of this code:
(a) Exchanges the first and third rows of A. ® l\1ultiplies each row of
A by its row number. (c) Multiplies the first t.hree columns of A by their column numbers.
(d) Multiplies the third row of A by 3, leaving the other rows un­
changed.
A9: Consider a linear system with two equations for two unknowns ::rl and
:C2. It is knowll that the syst.em has a single vector solution with com­
ponents Xl = 2 and .1:2 = 3. What is the reduced row echelon form of
the augmenteel matrix that represents the system?
r.
LO
b ; 2]
I
3 ­
AIO: For problem #9 above, circle the correct geometric interpretation of
t.he system:
e
two lines in the plane that intersect at a single point.
(b) two parallel but different lines in the plane.
(c) two identical lines in the plaHe.
(d) the intersection of two planes in three dimensional spR.('e.
3
Math 152 Test #1 version A
Part B - Long Answer Questions, 5 marks each
= (1 , 2,1), B = (2, - 2, 3) and C = (0, -3, 1) be three points in
three clirrwnsional spClce. Consider the triangle T whose vertices are A,
Band C,
B1: Let A
(a) [1 mark] What is t.he length of the side AB?
(b) [4] Find an equation of the plalle' P containing T in t.he form
(u:
+ by + ('z
=
d
with a, b, c, d determined.
AB=
b)
AL
=
p~
(0.-3,I)- (\,2,
V\l)~ n:: b
-
t)
Y.
- (-I, -t; 10).J::-/\
<!.
1'\
3'
-~
-I
-n·(~-A):a
-=-/
4
-~
~
2
t>
~V\l
~ .
~
Math 152 Test #1 version A
B2: Fiud all solutions (if any) to the system of equations
Xl
+ :1::3
+ X3
+X3
+'1:2
2Xl
3.1:1
I
0
3
2..
,
J
\
I
Yl +
(
1
0
~ ~ 1/3
(
\
0
Q
Y.. L
,
2
1
"2
+2X2
-4-
= 2
= 1
= 4
I
()
if
0
I
2.
'I2
3/,
(2) -2-(1)
-2
(3)-3(\) .
-2
(<.) +(-2.)
(3)+~("1.) .
~(~),,3h
5
' 2
-2 -I -3
-I
- ~/'2. 1 - Y2.
C) ~ (~)~2
I
)
~I
XL:: ~ _.1. :' '+/3 .
)(. ~ J/~
""
.
(g
Math 152 Test #1 version A
B3: COllsider t.he linear system of equations with parameters a and b:
2XI
Xl
+ 3X2
+ aX2
=
10
b
(a) [3 marks] For what value(s) of a and b does the system have a
unique solution?
(b) [1] For what value(s) of a and b does the system have an infinite
Humber of solutions?
(c) [1] For what value(s) of a and b does the system have no solutions?
3
Q
0..1
~~ fA,:#
hr-rb) LU~
",,\1
cvvt ~
t
~ ,'s
Gc
u~~ ~h~
b.
ll..; 2
2-,
b-=-S
~r ~ ~~'~f '
c)
6
~ i5