MATH 2107A
TEST 3
NOVEMBER 6, 2009
1 of 4
This test has two parts with a total of 30 marks. Part I has multiple choice questions, and
Part II has long answer questions. The test cannot be taken out from the examination
room. Only nonprogrammable calculators are allowed.
Duration: 50 minutes.
NAME (in ink):
STUDENT NO (in ink):
PART I: [6] Multiple choice questions. Circle the correct answer in ink.
[2] 1. Let
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
=
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
=
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
=
1
1
0
0
,
0
0
1
1
,
4
2
3
1
2
1
u
u
ν
and let
}
,
{
2
1
u
u
span
W
=
. If
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
=
d
c
b
a
proj
W
ν
,
What is the value of
d
?
a) 1
b) 2
c) 3
d) 4
e) 5
[2] 2. Let
A
be a
3
3
×
matrix such that
.
3
0
1
,
1
1
1
)
(
⎪
⎭
⎪
⎬
⎫
⎪
⎩
⎪
⎨
⎧
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
span
A
col
If
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
1
b
a
ν
is in
⊥
))
(
(
A
col
, What is the value of
b
a
+
?
a) -1
b) -3
c) 0
d) 1
e) 3
3.
Let
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
=
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
1
1
1
1
1
1
1
1
1
A
. You are given that
0
=
λ
is an eigenvalue of
A
.
What is the dimension of the corresponding eigenspace
0
E
?
a) 1
b) 2
c) 3
d) 4
e) 5
PART II: [24] Long answer questions. Show all your work.
[8] 1. Let
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
=
2
2
0
0
3
1
0
0
1
1
3
0
0
1
0
3
A
.
You are given that the eigenvalues of
A
are
3
,
1
−
=
λ
and 4.
Determine whether
A
is diagonalizable and if so, find an invertible matrix
P
and a
diagonal matrix
D
such that
1
−
=
PDP
A
.
I.
Eigenvectors for
1
−
=
λ
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
=
−
−
0
0
0
0
3
2
0
0
1
1
4
0
0
1
0
4
~
3
2
0
0
3
2
0
0
1
1
4
0
0
1
0
4
)
1
(
I
A
4
x
⇒
is a free variable.

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