Assignment 2 Solutions
1.
Determine, with proof, which of the following are subspaces of
R
3
and which are not.
a)
S
1
=
{
~x
∈
R
3

x
1

x
2
=
x
3
}
Solution: By deﬁnition
S
1
is a subset of
R
3
. Also,
0
0
0
∈
S
1
since 0

0 = 0. Thus,
S
1
is a
nonempty subset of
R
3
, so we an apply the Subspace Test.
Let
~x
=
x
1
x
2
x
3
,
~
y
=
y
1
y
2
y
3
∈
S
1
. Then
x
1

x
2
=
x
3
and
y
1

y
2
=
y
3
. This gives
~x
+
~
y
=
x
1
+
y
1
x
2
+
y
2
x
3
+
y
3
and
(
x
1
+
y
1
)

(
x
2
+
y
2
) =
x
1

x
2
+
y
1

y
2
=
x
3

y
3
So,
~x
+
~
y
satisﬁes the condition of
S
1
, so
~x
+
~
y
∈
S
1
.
Similarly,
c~x
=
cx
1
cx
2
cx
3
and
cx
1

cx
2
=
c
(
x
1

x
2
) =
cx
3
so
c~x
∈
S
1
.
Thus, by the Subspace Test,
S
1
is a subspace of
R
3
.
b)
S
2
=
1
a
b

a,b
∈
R
Solution: By deﬁnition,
S
2
is a subset of
R
3
, but
0
0
0
6∈
S
2
. Therefore,
S
2
is not a subspace.
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Winter '08
 All
 Math

Click to edit the document details