Math 152 - Linear Systems Test #2, Version B (MWF sections) Spring, 2010
University Of British Columbia
Name:
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ID Number:_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __
Instructions
• You should have seven pages including this cover.
• There are 2 parts to the test:
o Part A has 10 short questions worth 1 mark each
o Part Bhas 3 long questions worth 5 marks each
• Although all questions in each part are worth the same, some may be 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
Math 152 Test #2 version B
Part A - Short Answer Questions, 1 mark each
AI: Consider the matrices and vectors defined below:
Which multiplications below are well defined? (circle all that apply)
@
@
AC
BC
@)
A2: Consider the following lines of MATLAB code
°4]
[!
A = zeros(3,3);
5·, A(1,1)
4·,
A(2,3)
-1 ;
A(3,3)
0
0
-I
0
What matrix A will result?
A3: Write the transpose of the matrix below:
A=
[~89 ~
1'
-2
A4: Let
A
= [;
~]
. Is A invertible? (justify briefly).
~S I
rl.J)
A Ie o·
A5: Suppose the matrix P below is the transition matrix for a random walk.
Fill in the two missing (underlined) entries below
P =
[yg..
1/2]
3/4 '2
2
Math 152 Test #2 version B
For questions A6 and A7 below, consider the circuit shown below:
T~utt\~
,-'l/'iV'\r------::--.­
I
l
r)tt I
fl ~A V.
~_ ~_l l~~ ~
.1
r-«n~(a,vt~
2 S2­
4-Jt.
tb ~
{ OV
A6: Write a linear equation that describes the sum of the voltage drops
round loop 3 (the lower middle loop) in the circuit_ Your equation
should be in terms of the loop currents and voltages shown in the
diagram.
LI- l !>...
l L~ -l4-")
~ (L = 0­
A 7: Write a linear equation that expresses the current through the 5A cur­
rent source in terms of the loop currents in the diagram.
A8: Suppose that P is a projection, R a rotation and Q a reflection (all in
2D). Circle the statement below that is not true:
(a) p 2
=
P
(b) Q2 = 1
(c) RT
=
R- 1
T
~ Q=Q
(jJpT = p-l 3
i..ll..*
Math 152 Test #2 version B
A9: Let R be the matrix reresenting counterclockwise rotation by 7f /2 (90°)
in 2D. Find
R [:
1~ 1~41l
Q
t2cft ""f/~
=:
AID: What is the matrix representation of reflection through the lme y
in 2D followed by counter-clockwise rotation by 7f / 4 (45°)7
=
x
R·':: ~t 1i/4,
f~. h-r""""" \.ll ~
o
fZ=
~'. ,1f/,+ &t- (e~.' ;e..'~,. )
W~ tJ. ~ ~ Orcl...i'v~
~\rt d-t(}v\p f,o ~ 1t-t
~~.M"'-~ 0( ~.
o
(
r'L -1.].
! .
1
rtVLJI~~
tvt,1.) (J-rdJr
LV~
Wtf\....
Q is
J~~ f;~f.
4
Math 152 Test #2 version B
Part B - Long Answer Questions, 5 marks each
. Bl: Consider the matrix
A =
[
1 2 -2]
0 2 -1
-2 2 3
(a) [4 marks] Find A -\. Do the computations using Gaussian Elimi­
nation on an augmented matrix for full marks.
(b) [1] Solve
for x . H i nt: use your result from part (a)
- 1
O-}
o
i
I
o0
2
o
1_ -- I
-2
-
-
o
1
o
o
?
-'Iz
2
()
I
0
I
2.
-~
1
o
,
O·
'12. -lief ~
(]
o
t
!
J
o 0
o
o
o
, 2 -5/2
1/2
... V4
\ -)17..
o
~
~ (.L ~
s~t~..s:J..
~ wt
IA- i ,
~
0
o
'2
o
'I z.
o
L 0
I
-1
o
-2
' 1 00 1
[ ,~:
~
3
'
- ~/2
~ .
1/2
Math 152 Test #2 version B
B2: Let T be a linear transformation from three dimensions (3D) to 3D such.
that
T [
~ 1~
ln [n ~ [~1 1 ln ~ [~ 1
and
T
and
T
(a) [2 marks] Use the information above and the fact that T is linear
to determine
T[n
(b) [2J Now determine
~1
T [
(c) [lJ What is the matrix representation for T?
6
((.) T=
1
-2­
'2.
-I
-L
0
Math 152 Test #2 version B
B3: A model of the weather in Vancouver is constructed based purely on
probabilities. The weather on a given day is said to be either:
1. Sunny
2. Cloudy
3. or Rainy
The weather the next day is predicted based on the weather on the
current day. Suppose that the weather t~e next day will stay the same .
50% of the time, and change to the other types 25% each.
(a) [2 marks] Write the transition matrix for Vancouver weather based
on the model and numbering of states above.
(b) [2] If it is sunny one day, what is the chance that it will be ·sunny
two days later? Use matrix multiplication to work out this part
for full marks.
(c) [1] On average what fraction of the time will it be sunny in Van­
(CA)
rp ~
~/~ver v;rdi~to]thiS model? Justify briefly in words.
l
1ft
1/1
'iff
_
'1,"\ '14 'h
t (I) _
P!
(0)
'"
111.
If,..
1J4
.' 3/<6
-
i
~U
2~S,
(C) ~/V\tt
oJte.
~\~
~
I)
o€ 6e-.1Af;
~uv'\l\(1 ~
~~ ~ ~
~~
~tJM~t
~ ~ lK ~ tJ~
Iw VJt~ <; ~ bf. '/3 $\J"'.~ cJV\-,
~~t