56
CHAPTER 3.
SUBSPACES
Proof. (1)
A
0 = 0, so 0
∈
N
(
A
); (2) If
X, Y
∈
N
(
A
), then
AX
= 0 and
AY
= 0, so
A
(
X
+
Y
) =
AX
+
AY
= 0 + 0 = 0 and so
X
+
Y
∈
N
(
A
); (3)
If
X
∈
N
(
A
) and
t
∈
F
, then
A
(
tX
) =
t
(
AX
) =
t
0 = 0, so
tX
∈
N
(
A
).
For example, if
A
=
1
0
0
1
, then
N
(
A
) =
{
0
}
, the set consisting of
just the zero vector.
If
A
=
1
2
2
4
, then
N
(
A
) is the set of all scalar
multiples of [

2
,
1]
t
.
EXAMPLE 3.2.2
Let
X
1
, . . . , X
m
∈
F
n
.
Then the set consisting of all
linear combinations
x
1
X
1
+
· · ·
+
x
m
X
m
, where
x
1
, . . . , x
m
∈
F
, is a sub
space of
F
n
. This subspace is called the subspace
spanned
or
generated
by
X
1
, . . . , X
m
and is denoted here by
h
X
1
, . . . , X
m
i
. We also call
X
1
, . . . , X
m
a spanning family for
S
=
h
X
1
, . . . , X
m
i
.
Proof.
(1) 0 = 0
X
1
+
· · ·
+ 0
X
m
, so 0
∈ h
X
1
, . . . , X
m
i
; (2) If
X, Y
∈
h
X
1
, . . . , X
m
i
, then
X
=
x
1
X
1
+
· · ·
+
x
m
X
m
and
Y
=
y
1
X
1
+
· · ·
+
y
m
X
m
,
so
X
+
Y
=
(
x
1
X
1
+
· · ·
+
x
m
X
m
) + (
y
1
X
1
+
· · ·
+
y
m
X
m
)
=
(
x
1
+
y
1
)
X
1
+
· · ·
+ (
x
m
+
y
m
)
X
m
∈ h
X
1
, . . . , X
m
i
.
(3) If
X
∈ h
X
1
, . . . , X
m
i
and
t
∈
F
, then
X
=
x
1
X
1
+
· · ·
+
x
m
X
m
tX
=
t
(
x
1
X
1
+
· · ·
+
x
m
X
m
)
=
(
tx
1
)
X
1
+
· · ·
+ (
tx
m
)
X
m
∈ h
X
1
, . . . , X
m
i
.
For example, if
A
∈
M
m
×
n
(
F
), the subspace generated by the columns of
A
is an important subspace of
F
m
and is called the
column space
of
A
. The
column space of
A
is denoted here by
C
(
A
). Also the subspace generated
by the rows of
A
is a subspace of
F
n
and is called the
row space
of
A
and is
denoted by
R
(
A
).
EXAMPLE 3.2.3
For example
F
n
=
h
E
1
, . . . , E
n
i
, where
E
1
, . . . , E
n
are
the
n
–dimensional unit vectors. For if
X
= [
x
1
, . . . , x
n
]
t
∈
F
n
, then
X
=
x
1
E
1
+
· · ·
+
x
n
E
n
.
EXAMPLE 3.2.4
Find a spanning family for the subspace
S
of
R
3
defined
by the equation 2
x

3
y
+ 5
z
= 0.