-
A vector space consists of the following four
elements:
-
A set of vectors
VV,
-
A field of scalars
(where, for our purposes,
is either
R or
C),
-
The operations of vector addition "+"
(i.e., + :
V×V→V
V
V
V
)
-
The operation of scalar multiplication
"⋅"(i.e., ⋅ :
×V→V
V
V
)
Table 1
| Properties |
Examples |
| commutativity |
x⇀+y⇀=y⇀+x⇀
x
y
y
x
|
| associativity |
(x⇀+y⇀)+z⇀=x⇀+(y⇀+z⇀)
x
y
z
x
y
z
|
|
(αβ)x⇀=α(βx⇀)
α
β
x
α
β
x
|
| distributivity |
α⋅(x⇀+y⇀)=(α⋅x⇀)+(α⋅y⇀)
α
⋅
x
y
⋅
α
x
⋅
α
y
|
|
(α+β)x⇀=αx⇀+βx⇀
α
β
x
α
x
β
x
|
| additive identity |
∃
0
,0∈V:x⇀+0=x⇀ ,
x⇀∈V
x
V
0
0
V
x
0
x
|
| additive inverse |
∃
−x⇀
,(−x⇀)∈V:x⇀+−x⇀=0 ,
x⇀∈V
x
V
x
x
V
x
x
0
|
| multiplicative identity |
1⋅x⇀=x⇀ ,
x⇀∈V
x
V
⋅
1
x
x
|
Important examples of vector spaces include
Table 2
|
Properties
|
Examples
|
|
real NN-vectors
|
V=RN
V
N
,
=R
|
|
complex NN-vectors
|
V=CN
V
N
,
=C
|
|
sequences in
"
lp
lp
"
|
V=
xn
∃n∈Z:∑n=−∞∞|xn|p<∞
V
x
n
n
n
x
n
p
,
=C
|
|
functions in "
ℒp
ℒp
"
|
V=
ft
∫−∞∞|ft|pdt<∞
V
f
t
t
f
t
p
,
=C
|
where we have assumed the usual definitions of addition and
multiplication. From now on, we will denote the arbitrary
vector space (
VV,
, +, ⋅) by the
shorthand
VV and
assume the usual selection of
(
, +, ⋅). We will
also suppress the "⋅" in scalar multiplication, so that
α⋅x
⋅
α
x
becomes
αx
α
x
.
-
A subspace of VV is a subset
M⊂V
M
V
for which
-
(x⇀+y⇀)∈M ,
x⇀∈M∧y⇀∈M
x
y
x
M
y
M
x
y
M
-
αx⇀∈M ,
x⇀∈M∧α∈
x
M
α
α
x
M
Note that every subspace must contain
00,
and that VV is a
subspace of itself.
-
The span of set
S⊂V
S
V
is the subspace of VV containing all linear
combinations of vectors in SS. When
S=x⇀0…x⇀
N
-
1
S
x0
…
x
N
-
1
,
spanS≔
∑
i
=0N−1
α
i
x
i
α
i
∈
≔
span
S
i
0
N
1
α
i
x
i
α
i
-
A subset of linearly-independent vectors
x⇀0…x⇀
N
-
1
⊂V
x0
…
x
N
-
1
V
is called a basis for V when its span
equals VV. In such a
case, we say that VV
has dimension NN.
We say that VV is
infinite-dimensional if it contains an infinite number
of linearly independent vectors.
-
VV is a direct
sum of two subspaces MM and NN, written
V=M⊕N
V
M
N
, iff every
x⇀∈V
x
V
has a unique representation
x⇀=m⇀+n⇀
x
m
n
for
m⇀∈M
m
M
and
n⇀∈N
n
N
.
Note that this requires
M∩N=0
M
N
0
