Math 136
Assignment 7 Solutions
1.
Show the each of the following sets form a basis for the subspace that they span, and
determine the coordinates of
±x
and
±
y
with respect to the basis.
a)
{
(1
,
1
,
0
,
1
,
0)
,
(1
,
0
,
2
,
1
,
1)
,
(0
,
0
,
1
,
1
,
3)
}
;
= (2
,

2
,
5
,

1
,

5),
±
y
=(

1
,

3
,
3
,

2
,

1).
Consider
0=
c
1
(1
,
1
,
0
,
1
,
0) +
c
2
(1
,
0
,
2
,
1
,
1) +
c
3
((0
,
0
,
1
,
1
,
3)
c
1
+
c
2
,c
1
,
2
c
2
+
c
3
1
+
c
2
+
c
3
2
+3
c
3
)
We rowreduce the coeFcient matrix to get
1 1 0
1 0 0
0 2 1
1 1 1
0 1 3
∼
1 0 0
0 1 0
0 0 1
0 0 0
0 0 0
.
Thus, the only solution is
c
1
=
c
2
=
c
3
= 0 so the set of vectors is linearly independent and
so the vectors in
B
form a basis for the subspace which they span.
The coordinates of
with respect to
B
and the coordinates of
±
y
with respect to
B
are
determined by rowreducing the augmented systems
c
1
(1
,
1
,
0
,
1
,
0) +
c
2
(1
,
0
,
2
,
1
,
1) +
c
3
(0
,
0
,
1
,
1
,
3) = (2
,

2
,
5
,

1
,

5)
d
1
(1
,
1
,
0
,
1
,
0) +
d
2
(1
,
0
,
2
,
1
,
1) +
d
3
(0
,
0
,
1
,
1
,
3) = (

1
,

3
,
3
,

2
,

1)
.
We make one doubly augmented matrix and rowreduce to get
1 1 0
2

1
1 0 0

2

3
0 2 1
53
1 1 1

1

2
0 1 3

5

1
∼
1 0 0

2

3
0 1 0
42
0 0 1

3

1
0 0 0
00
0 0 0
Thus [
]
B
=
c
1
c
2
c
3
=

2
4

3
and [
±
y
]
B
=
d
1
d
2
d
3
=

3
2

1
.
b)
±²
11
10
³
,
²
01
³
,
²
20
0

1
³´
;
=
²
12
³
,
±
y
=
²

41
14
³
.
Solution: Consider
²
³
=
c
1
²
³
+
c
2
²
³
+
c
3
²
0

1
³
=
²
c
1
+2
c
3
c
1
+
c
2
c
1
+
c
2
c
2

c
3
³
.
1