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Definition (Vector Space)
Let F be a field. A set V with two binary operations: + (addition) and
×
{\displaystyle \times }
Vector Space if it has the following properties:
(
V
,
+
)
{\displaystyle (V,+)}
(
a
+
b
)
v
=
a
v
+
b
v
{\displaystyle (a+b)v=av+bv}
a
,
b
∈
F
{\displaystyle a,b\in F}
v
∈
V
{\displaystyle v\in V}
a
(
v
+
u
)
=
a
v
+
a
u
{\displaystyle a(v+u)=av+au}
a
∈
F
{\displaystyle a\in F}
v
,
u
∈
V
{\displaystyle v,u\in V}
(
a
b
)
v
=
a
(
b
v
)
{\displaystyle (ab)v=a(bv)}
1
F
v
=
v
{\displaystyle 1_{F}v=v}
The scalar multiplication is formally defined by
F
×
V
→
ϕ
V
{\displaystyle F\times V{\xrightarrow {\phi }}V}
ϕ
(
(
f
,
v
)
)
=
f
v
∈
V
{\displaystyle \phi ((f,v))=fv\in V}
Elements in F are called scalars, while elements in V are called vectors.
Some Properties of Vector Spaces
0
F
v
=
0
V
=
a
0
V
{\displaystyle 0_{F}v=0_{V}=a0_{V}}
(
−
1
F
)
v
=
−
v
{\displaystyle (-1_{F})v=-v}
a
v
=
0
⟺
a
=
0
or
v
=
0
{\displaystyle av=0\iff a=0{\text{ or }}v=0}
Proofs:
0
F
v
=
(
0
F
+
0
F
)
v
=
0
F
v
+
0
F
v
⇒
0
V
=
0
F
v
.
A
l
s
o
,
a
0
V
=
a
(
0
V
+
0
V
)
=
a
0
V
+
a
0
V
⇒
a
0
V
=
0
V
{\displaystyle 0_{F}v=(0_{F}+0_{F})v=0_{F}v+0_{F}v\Rightarrow 0_{V}=0_{F}v.Also,a0_{V}=a(0_{V}+0_{V})=a0_{V}+a0_{V}\Rightarrow a0_{V}=0_{V}}
We want to show that
v
+
(
−
1
F
)
v
=
0
V
{\displaystyle v+(-1_{F})v=0_{V}}
v
+
(
−
1
F
)
v
=
1
F
v
+
(
−
1
F
)
v
=
(
1
F
+
(
−
1
F
)
)
v
=
0
F
v
=
0
V
{\displaystyle v+(-1_{F})v=1_{F}v+(-1_{F})v=(1_{F}+(-1_{F}))v=0_{F}v=0_{V}}
Suppose
a
v
=
0
{\displaystyle av=0}
a
≠
0
{\displaystyle a\neq 0}
a
−
1
(
a
v
)
=
a
−
1
0
=
0
⇒
1
F
v
=
v
=
0
{\displaystyle a^{-1}(av)=a^{-1}0=0\Rightarrow 1_{F}v=v=0}
