Math 152 Test #2 version B
Part A - Short Answer Questions, 1 mark each
For questions A1, A2 and A3 consider the resistor network shown below.
The resistances and sources are known. The voltage drops across the current
sources are E 1 and E 2 respectively.
Al: A linear system is set up for this network using the loop current technique. List the unknowns in this system. You do not need to write the
equations in the system.
~I
.
t1..J
.
l~ 1 t, l E'2
A2: Write the linear equation that describes voltage drops around loop 3
(that is, the loop that corresponds to the loop current i 3 ) in the network
above.
R! ( i3- ~) -t ~~ ( \-; -t ~l -\11. ~o .
A3: Write the linear equation that matches the loop currents to the first
current source.
(\
A4: Let a
=
l~J
-~1. ==
I1 .
[1 2]. Write the result of the matrix product aT a .
[I
t;J -""
2
Math 152 Test #2 version B
A5: Consider the following lines of MATLAB code:
f"l
~ ::..
\1
Q
·
A= zeros(3,3);
3 0'l ~for AC1,1)
i~1:2
"].. ~
f:>O~
.
=
L!J
2;
A(i, i+1)=3;
end
A(3 , 3) = 5;
A(: ,3)
Write the output from the final line above.
A6: Let b = [3 , 9, 5, 3]. Either write b as a linear combination of [1, 2, 1, 2],
[3, 3, 3, 3], and [1 , 2, 3, 4] or show it cannot be done.
3
\ ~ 3
2 3 2
'
~
~ ~
""
3 3 ( S'
tȣ-t
0
0
3
I
-~
0
0
~
1'3
3
2
0 - 3 2 -3
I~
}
~~
A 7: A certain car from the car-share company Go2Car can be parked at
three different locations: UBC, Downtown, Richmond. No matter
where the car is parked on a given day, 1/2 of the time it will be
returned to the same place the next day. If the car is parked at UBC
on a given day, then 1/3 of the time it will be parked Downtown the
next day. If the car is parked in Richmond, it will never be Downtown
the next day. If it is parked Downtown, it is equally likely to be at UBC
or Richmond the next day. Write the probability transition matrix for
the car location, using the ordering: UBC, Downtown, Richmond.
1
ft
0
\{~
3
V\.-0
s~~
Math 152 Test #2 version B
A8: Consider the 2D projection matrix
4/5 -2/5 ]
[ -2/5 1/5
.
Write a vector in the direction of the line that this matrix projects on
to.
l'l, -!J
A9: Find the determinant of
2
lA=
I
I I
-L- 0
I
31
-z
L+
5
2
-3 -3
1/2 1
1
2
~(8::.
~ ~
s
I
I
I
I
-I -1
I
'l
-~
\
t
tf ~
-2.(-loJ +2.(-2.3)
/A:: -Co
+j 3 = - ~.
"'2., 7
AlO: A random walk with two states has a probability transition matrix P.
It is known that
p2 = [ 5/8 3/8 ]
3/8 5/8 .
Determine two possible matrices P that satisfy the condition above.
~/q.]
'lq. .
Sft+
l
lf£1
4
=-(o .
Math 152 Test #2 version B
B2: Let SandT be transformations such that
(a) [1 mark] Write the matrix representation of S .
•
(b) [1] Write the matrix representation ofT o S. Note: depending on
how you approach this problem, you may want to return to it after
part (d) below.
(c) [1] Write e 1 = [
~ ] as a linear combination of [ ~3 ] and [ ~2 ] .
· (d) [2] Write the matrix representation ofT.
Math 152 Test #2 version B
B3: Consider two lines:
√
l1 : y = x/ 3 (makes an angle π/6 with the positive x-axis)
√
l2 : y = 3x (makes an angle π/3 with the positive x-axis)
(a) [1 mark] Find the matrix representation of the linear transformation Ref1 , which is a reflection relative to the line l1 .
(b) [1] Find the matrix representation of the linear transformation
Ref2 , which is a reflection relative to the line l2 .
(c) [1] Show using matrix multiplication that the application of the reflection Ref1 twice gives the original vector, i.e. that Ref1 Ref1 a =
a for any vector a.
(d) [1] For any a, consider b = Ref1 a and c = Ref2 a as in the figure
below. It can be shown that the resulting vectors b and c are
related by a linear transformation T , as c = T b. Find the matrix
representation of T .
(e) [1] T in (d) is a rotation. Find the angle of the rotation.
y
l2
a
c
l1
x
b
An extra blank page is added next for your work on this question
7
Version B, problem B3:
• (a) [1 mark] The first line makes an angle θ1 = π/6 with the x axis, so
√ cos(2θ1 ) sin(2θ1 )
cos(π/3) sin(π/3)
1/2
3/2
√
Ref1 =
=
=
.
sin(2θ1 ) − cos(2θ1 )
sin(π/3) − cos(π/3)
3/2 −1/2
• (b) [1 mark] The second line makes an angle θ2 = π/3 with the x axis, so
√
cos(2θ2 ) sin(2θ2 )
cos(2π/3) sin(2π/3)
−1/2
3/2
√
Ref2 =
=
=
.
sin(2θ2 ) − cos(2θ2 )
sin(2π/3) − cos(2π/3)
3/2 1/2
• (c) [1 mark]
√
√
1/2
3/2 √1/2
3/2
1 0
√
a = a.
Ref1 Ref1 a =
a=
0 1
3/2 −1/2
3/2 −1/2
• (d) [1 mark] Since b = Ref1 a, then
Ref1 b = Ref1 Ref1 a = a. So, a = Ref1 b. Since
c = Ref2 a, then c = Ref2 Ref1 b . This implies that
√
√ √ −1/2
3/2
1/2
3/2
1/2
−
3/2
√
T = Ref2 Ref1 = √
= √
3/2 1/2
3/2 −1/2
3/2
1/2
• (e) [1 mark] T is a rotation matrix, so
√
1/2
−
cos(θ) − sin(θ)
3/2
= √
Rotθ =
.
sin(θ) cos(θ)
3/2
1/2
It is clear that θ = π/3 since sin(π/3) =
√
3/2 and cos(π/3) = 1/2.