MATH 211 (Siefken): Homework #4
Due June 27, 2011
Question #1
(a) Consider the set of vectors
S
=
{
v
1
,
v
2
,
v
3
,
v
4
,
v
5
,
v
6
}
where
v
1
=
1
2
3
4
v
2
=

2
4
4
6
v
3
=

3
2
1
2
v
4
=
0
10
8
10
v
5
=
6
6
6
6
v
6
=

7
0
1
4
.
(a) Find two distinct, maximal, linearly independent subsets of
S
.
If
A
= [
v
1

v
2

v
3

v
4

v
5

v
6
] we compute that
rref(
A
) =
1
0

1
0
2

1
0
1
1
0

2
3
0
0
0
1
1

1
0
0
0
0
0
0
.
We see that the rank of
A
is three and so the largest set of linearly independent vectors among the columns
of
A
has size 3. The pivots of the rref form of
A
indicate a maximal linearly independent set, and so we may
quickly list one such set
S
1
=
{
v
1
,
v
2
,
v
4
}
.
The rref form also shows all dependence relations among the columns of
A
.
Thus we see that the sets
{
v
1
,
v
2
}
,
{
v
1
,
v
3
}
,
{
v
2
,
v
3
}
are sets of linearly independent vectors. Further, each of these sets is linearly
independent from
v
4
and so we can find a different maximal set of linearly independent vectors,
S
2
=
{
v
1
,
v
3
,
v
4
}
.
There are many, many other maximal sets of linearly independent vectors, all of which can be obtained by
analyzing the dependence relations.
Alternatively,
one could scramble the columns of
A
and row reduce again.
For example if
B
=
[
v
6

v
3

v
4

v
1

v
5

v
2
], rref(
B
) indicates that
S
3
=
{
v
3
,
v
4
,
v
6
}
is a maximal linearly independent set.
(b) What can you say about the span of the sets in part (a)? What is dim(span(
S
))?
We know that a maximal linearly independent subset of
S
is a basis for span(
S
). Thus span(
S
1
) = span(
S
2
) =
span(
S
). Since there are three vectors in this basis, dim(span(
S
)) = 3.
(c) If
A
= [
v
1

v
2

v
3

v
4

v
5

v
6
], what is dim(null(
A
)) and dim(null(
A
T
))?
Computed in part (a), we know rank(
A
) = 3. Since dim(row(
A
)) + dim(null(
A
)) = 6 and dim(row(
A
)) = 3,
we know dim(null(
A
)) = 3. Similarly, we know dim(col(
A
)) + dim(null(
A
T
)) = 4 and dim(col(
A
)) = 3 and
so dim(null(
A
T
)) = 1.
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 Summer '10
 SOSPEDRAALFONSO
 Linear Algebra, Algebra, Vectors, Vector Space, basis, linearly independent vectors, Siefken

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