3.1/3.2 DEFINITONS AND EXAMPLES OF VECTOR SPACES
AND SUBSPACES
KIAM HEONG KWA
p. 113 A (real)
linear space
or a (real)
vector space
is an ordered triple
(
V,
⊕
,
±
)
, where
V
is a nonempty set and
⊕
:
V
×
V
→
V
and
±
:
R
×
V
→
V
are functions, such that the following axioms hold:
A1.
p
⊕
q
=
q
⊕
p
for any
p
,
q
∈
V
.
A2.
(
p
⊕
q
)
⊕
r
=
p
⊕
(
q
⊕
r
)
for any
p
,
q
,
r
∈
V
. (In view of
this, we write
p
⊕
q
⊕
r
=
p
⊕
(
q
⊕
r
)
.)
A3. There is an element
0
∈
V
such that
p
⊕
0
=
p
for any
p
∈
V
.
A4. For each
p
∈
V
, there is an element

p
∈
V
such that
p
⊕
(

p
) =
0
.
A5.
α
±
(
p
⊕
q
) = (
α
±
p
)
⊕
(
α
±
q
)
for any
α
∈
R
and any
p
,
q
∈
V
.
A6.
(
α
+
β
)
±
p
= (
α
±
p
)
⊕
(
β
±
p
)
for any
α,β
∈
R
and any
p
∈
V
.
A7.
(
αβ
)
±
p
=
α
±
(
β
±
p
)
for any
α,β
∈
R
and any
p
∈
V
.
A8.
1
±
p
=
p
for any
p
∈
V
.
Notation: Here we have written
p
⊕
q
for
⊕
(
p
,
q
)
and
α
±
p
for
±
(
α,
p
)
for any
α
∈
R
and any
p
,
q
∈
V
.
Implicit in the deﬁnition of the vector space
(
V,
⊕
,
±
)
are the
clo
sure properties
:
C1. For any scalar
α
and
p
∈
V
,
α
±
p
∈
V
.
C2. For any
p
,
q
∈
V
,
p
⊕
q
∈
V
.
The functions
⊕
and
±
are called
addition
and
scalar multiplica
tion
respectively. The elements of
V
are called
vectors
. In this
context, the elements of
R
are called
scalars
. It is customary to re
fer to
V
as the linear space instead of the ordered triple
(
V,
⊕
,
±
)
on
the assumption that the addition and scalar multiplication are tacitly
given.
Notation: It is not uncommon to write
+
for
⊕
, so that
p
⊕
q
is
written as
p
+
q
for any
p
,
q
∈
V
. It is also not uncommon to
represent
±
by juxtaposition, so that
α
±
p
is written as
α
p
for any
α
∈
R
and any
p
∈
V
.
Date
: June 26, 2011.
1