Math 3013
Homework Set 4
Problems from
§
1.6 (pgs. 99101 of text): 1,3,5,7,9,11,17,19,21,24,26,35,37,38
1. (Problems 1.6.1, 1.6.3, 1.6.4, 1.6.7, 1.6.9 in text). Determine whether the indicated subset is a subspace
of the given
R
n
.
(a)
W
=
{
[
r,

r
]

r
∈
R
}
in
R
2
(b)
W
=
{
[
n,m
]

n
and
n
are integers
}
in
R
2
(c)
W
=
{
[
x,y,z
]

x,y,z
∈
R
and
z
= 3
x
+ 2
}
in
R
3
(d)
W
=
{
[
x,y,z
]

x,y,z
∈
R
and
z
= 1,
y
= 2
x
}
in
R
3
(e)
W
=
{
[2
x
1
,
3
x
2
,
4
x
3
,
5
x
4
]

x
i
∈
R
}
in
R
4
2. (Problem 1.6.11 in text). Prove that the line
y
=
mx
is a subspace of
R
2
. (Hint: write the line as
W
=
{
[
x,mx
]

x
∈
R
}
.)
3. (Problems 1.6.17, 1.6.19 and 1.6.21 in text). Find a basis for the solution set of the following homogeneous
linear systems.
(a)
3
x
1
+
x
2
+
x
3
=
0
6
x
1
+ 2
x
2
+ 2
x
3
=
0

9
x
1

3
x
2

3
x
3
=
0
(b)
2
x
1
+
x
2
+
x
3
+
x
4
=
0
x
1

6
x
2
+
x
3
=
0
3
x
1

5
x
2
+ 2
x
3
+
x
4
=
0
5
x
1

4
x
2
+ 3
x
3
+ 2
x
4
=
0
(c)
x
1

x
2
+ 6
x
3
+
x
4

x
5
=
0
3
x
1
+ 2
x
2

3