MATH 136 Winter 2008 Midterm Examination: Solutions
February 13, 2008
[10]
Question 1:
Mark each statement True or False. No justiﬁcation necessary.
1 mark for each correct answer,

1
2
for each wrong answer and
0
for no answer.
(i) Let
A
be a 4
×
3 real matrix. Then Col
A
is a subspace of
R
4
.
True
√
±
False
±
(ii) If
A
is any 5
×
3 matrix and
~
b
is any vector in
R
5
such that the equation
A~x
=
~
b
has a solution,
then the solution is unique.
True
±
False
√
±
(iii) Let
V
be a vector space over a ﬁeld
F
. If
a~x
=
b~x
for
a, b
∈
F
and
~x
∈
V
, then
a
=
b
.
True
±
False
√
±
(iv) If
f
and
g
are polynomials of degree 100 over rationals, then
f
+
g
is a polynomial of degree 100.
True
±
False
√
±
(v) The span of three vectors in a vector space
V
is a subspace of
V
.
True
√
±
False
±
(vi) The set of real numbers is a vector space over the ﬁeld of rational numbers.
True
√
±
False
±
(vii) Let
V
=
C
2
=
{
(
x, y
) :
x, y
∈
C
}
and
W
=
{
(
x, y
)
∈
V
:
x
2
+
y
2
= 0
}
. Then
W
is a subspace of
V
.
True
±
False
√
±
~v
1
= (1
, i
)
, ~v
2
= (

1
, i
) gives
~v
1
+
~v
2
= (0
,
2
i
)
/
∈
W
.
(viii) Let
T
:
R
2
→
R
2
be given by
T
(
u
1
, u
2
) =
±
(
u
1
, u
2
)
if
u
1
6
= 0
(0
,
0)
if
u
1
= 0
. Then
T
is a linear transformation.
True
±
False
√
±
~v
1
= (0
,
1)
, ~v
2
= (1
,
1) gives
T
(
~v
1
+
~v
2
)
6
=
T
(
~v
1
) +
T
(
~v
2
).
(ix) Let
T
:
R
3
→
R
be given by
T
(
u
1
, u
2
, u
3
) =
u
1
+
u
2
+
u
3
. Then
T
is a linear transformation.
True
√
±
False
±
(x) Let
T
:
R
n
→
R
be such that
T
(
~u
) =largest component of
~u
. Then
T
is a linear transformation.
True
±
False
√
±
~v
1
= (1
,
2
,
···
, n
)
,~v
2
= (

1
,

2
,
···
,

n
) gives
T
(
~v
1
) =
n, T
(
~v
2
) =

1 but
T
(
~v
1
+
~v
2
) = 0.