MATH 110: LINEAR ALGEBRA
SPRING 2007/08
PROBLEM SET 1 SOLUTIONS
1.
Prove that the following are vector spaces over
R
:
(a) polynomials of degree not more than
d
,
P
d
=
{
a
0
+
a
1
x
+
· · ·
+
a
d
x
d

a
i
∈
R
for all
i
}
,
(b)
m
by
n
matrices
R
m
×
n
=
{
[
a
ij
]
m,n
i,j
=1

a
ij
∈
R
for all
i, j
}
.
The addition and scalar multiplication operations for polynomials and matrices are as defined
in the lectures.
Solution.
Routine.
2.
Let
V
be a vector space over
R
with addition and scalar multiplication denoted by + and
·
respectively. Let
W
=
V
×
V
=
{
(
v
1
,
v
2
)

v
1
,
v
2
∈
V
}
. Prove that
W
is a vector space over
C
with addition defined by
(
u
1
,
u
2
)
(
v
1
,
v
2
) = (
u
1
+
v
1
,
u
2
+
v
2
)
for all (
u
1
,
u
2
)
,
(
v
1
,
v
2
)
∈
W
and scalar multiplication defined by
(
a
+
bi
)
(
v
1
,
v
2
) = (
a
·
v
1

b
·
v
2
, b
·
v
1
+
a
·
v
2
)
for all
a
+
bi
∈
C
and (
v
1
,
v
2
)
∈
W
. Here
i
=
√

1 and
a, b
∈
R
.
Solution.
Routine.
3.
Which of the following are subspaces of
R
2
? Justify your answers.
(a)
U
a
=
{
(
x, y
)
∈
R
2

x
2
+
y
2
= 0
, x, y
∈
R
}
,
(b)
U
b
=
{
(
x, y
)
∈
R
2

x
2

y
2
= 0
, x, y
∈
R
}
,
(c)
U
c
=
{
(
x, y
)
∈
R
2

x
2

y
= 0
, x, y
∈
R
}
,
(d)
U
d
=
{
(
x, y
)
∈
R
2

x

y
= 0
, x, y
∈
R
}
,
(e)
U
e
=
{
(
x, y
)
∈
R
2

x

y
= 1
, x, y
∈
R
}
.
If we replace
R
by
C
and
R
2
by
C
2
above, will any of your answers change?
Solution.
Note that
U
a
=
{
(0
,
0)
}
and so is a subspace. Let
α, β
∈
R
.
U
d
is a subspace by
Theorem
1.8
: if
x
1

y
1
= 0 and
x
2

y
2
= 0, then (
αx
1
+
βx
2
)

(
αy
1
+
βy
2
) =
α
(
x
1

y
1
) +
β
(
x
2

y
2
) = 0. (0
,