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MA 265 LECTURE NOTES: FRIDAY, MARCH 21
Row and Column Space
Rank of a Matrix.
Let
A
be an
m
×
n
matrix. We have seen that the row rank of
A
is the dimension
of the subspace of
R
n
spanned by the rows of
A
; and that the column space of
A
is the dimension of the
subspace of
R
m
spanned by the columns of
A
. We show that the row rank is equal to the column rank.
To be explicit, let’s write
A
=
a
11
a
12
···
a
1
n
a
21
a
22
···
a
2
n
.
.
.
.
.
.
.
.
.
.
.
.
a
m
1
a
m
2
···
a
mn
.
and denote the
i
th row and the
j
th column of
A
as the matrices
row
i
(
A
) =
±
a
i
1
a
i
2
···
a
in
²
∈
R
n
and
col
j
(
A
) =
a
1
j
a
2
j
.
.
.
a
mj
∈
R
m
.
Let
r
be the row rank of
A
, and say that
S
=
{
w
1
,
w
2
, ...,
w
r
}
is a basis for the row space of
A
. Then
each row can be expressed as a linear combination of these basis vectors:
row
1
(
A
)
=
c
11
w
1
+
c
12
w
2
+
···
+
c
1
r
w
r
row
2
(
A
)
=
c
21
w
1
+
c
22
w
2
+
···
+
c
2
r
w
r
.
.
.
row
m
(
A
) =
c
m
1
w
1
+
c
m
2
w
2
+
···
+
c
mr
w
r
Upon writing
w
i
=
±
b
i
1
b
i
2
···
b
in
²
and looking at the
j
th columns, we have the following expressions:
a
ij
=
c
11
b
1
j
+
c
12
b
2
j
+
···
+
c
1
r
b
rj
a
2
j
=
c
21
b
1
j
+
c
22
b
2
j
+
···
+
c
2
r
b
rj
.
.
.
a
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