Solutions of Selected Problems in HW5
3.2.10
(b)
{
(1
,
0
,
0)
,
(0
,
1
,
1)
,
(1
,
0
,
1)
,
(1
,
2
,
3)
}
We have to check existence of
C
1
,
· · ·
, C
4
such that
C
1
1
0
0
+
C
2
0
1
1
+
C
3
1
0
1
+
C
4
1
2
3
=
x
1
x
2
x
3
for arbitrary
x
1
, x
2
, x
3
.
This gives the augmented matrix
1
0
1
1
x
1
0
1
0
2
x
2
0
1
1
3
x
3
and its RREF is
1
0
0
0
x
1
+
x
2

x
3
0
1
0
2
x
2
0
0
1
1

x
2
+
x
3
.
This system is consistent and always has a solution no matter what
x
1
, x
2
and
x
3
. Thus
it spans
R
3
.
(c)
{
(2
,
1
,

2)
,
(3
,
2
,

2)
,
(2
,
2
,
0)
}
The corresponding augmented matrix is
2
3
2
x
1
1
2
2
x
2

2

2
0
x
3
and its RREF is
1
0

2
2
x
1

3
x
2
0
1
2

x
1
+ 2
x
2
0
0
0
2
x
1

2
x
2
+
x
3
.
The system is inconsistent if 2
x
1

2
x
2
+
x
3
6
= 0. This implies that no linear combination
of the above three vectors can make a vector (
x
1
, x
2
, x
3
) with 2
x
1

2
x
2
+
x
3
6
= 0.
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 Spring '08
 KIM
 Linear Algebra, Algebra, Vector Space, CN, augmented matrix

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