Harvard University
Solutions
Midterm 1 for Math 121, Fall 2007
Monday, October 22, 2007
1.
(20 points)
a)
True
b)
False
c)
False
d)
True
e)
True
2.
(40 points)
a)
Clearly, the zero vector lies in each of the subsets in (i), (ii). Thus, check if for any two
vectors
v
,
w
from the subset and for any
λ
∈
R
the vector
v
+
λw
is in the subset as well.
(i)
{
(0
,x,
2
x,
3
x
)

x
∈
R
}
is a subspace of
R
4
.
Proof: Let
v
= (0
,x,
2
x,
3
x
) and
w
= (0
,y,
2
y,
3
y
) be two arbitrary vectors from this
subset. Then it is
v
+
λw
= (0
, x
+
λy,
2
x
+ 2
λy,
3
x
+ 3
λy
) = (0
,z,
2
z,
3
z
)
with
z
=
x
+
λy
. Thus,
v
+
λw
is in the subset. This proves that this subset is indeed a
subspace of
R
4
.
(ii)
{
(
x
4
,x
3
,x
2
,x
)

x
∈
R
}
is not a subspace of
R
4
?
Proof: Consider the vector
v
= (1
,
1
,
1
,
1), which belongs to this subset. The vector
2
v
= (2
,
2
,
2
,
2) is not in this subset, as there does not exist any
x
∈
R
with (2
,
2
,
2
,
2) =
(
x
4
,x
3
,x
2
,x
). This proves that this subset is not a subspace of