Partial Solution Set, Leon Section 5.2
5.2.1
For each of the following matrices, determine bases for the four fundamental subspaces.
(a)
A
=
±
3 4
6 8
²
. The rowechelon form of
A
is
±
3 4
0 0
²
, from which we obtain three
of the four bases:
(a) For
N
(
A
), we can use
{
(4
,

3)
T
}
.
(b) For
R
(
A
T
), we can use
{
(3
,
4)
T
}
.
(c) For
R
(
A
), we can use
{
(1
,
2)
T
}
.
To ﬁnd a basis for
N
(
A
T
), we either use elimination on
A
T
or take advantage of
what we’ve learned about orthogonal subspaces by now. In either case, we obtain
something very much like
{
(2
,

1)
T
}
.
(c) The matrix is
A
=
4

2
1
3
2
1
3
4
. As in (a), we ﬁrst ﬁnd the rowechelon form of
A
,
which is
U
=
4

2
0 7
/
2
0
0
0
0
. The bases:
(a) For
N
(
A
),
{
0
}
.
(b) For
R
(
A
T
), we can use
{
(4
,

2)
T
,
(1
,
3)
T
}
.
(c) For
R
(
A
), we can use
{
(4
,
1
,
2
,
3)
T
,
(

2
,
3
,
1
,
4)
T
}
.
After elimination on