Chapter 4
Vector Spaces Associated with
Matrices
Section 4.1
Row Spaces and Column Spaces
Discussion 4.1.1
Each
m
×
n
matrix is naturally associated with three vector spaces,
namely, the row space, the column space and the nullspace. These three vector spaces
provide us with insights into the relationships between solutions of linear systems and
the coefficient matrix. In the following, we shall first discuss the row and column spaces
of a matrix.
Definition 4.1.2
Let
A
= (
a
ij
) be an
m
×
n
matrix, i.e.
A
=
a
11
a
12
· · ·
a
1
n
a
21
a
22
· · ·
a
2
n
.
.
.
.
.
.
.
.
.
a
m
1
a
m
2
· · ·
a
mn
.
The
row space
of
A
is the subspace of
R
n
spanned by the rows of
A
. The
column space
of
A
is the subspace of
R
m
spanned by the columns of
A
. Let
r
1
,
r
2
,
· · ·
,
r
m
be the
m
rows of
A
, i.e.
r
1
= (
a
11
, a
12
, . . . , a
1
n
)
,
r
2
= (
a
21
, a
22
, . . . , a
2
n
)
,
.
.
.
r
m
= (
a
m
1
, a
m
2
, . . . , a
mn
)
,
and let
c
1
,
c
2
,
· · ·
,
c
n
be the
n
columns, i.e.