# Abstract Algebra/Vector Spaces

## DefinitionEdit

Definition (Vector Space)
Let F be a field. A set V with two binary operations: + (addition) and ${\displaystyle \times }$  (scalar multiplication), is called a Vector Space if it has the following properties:
1. ${\displaystyle (V,+)}$  forms an abelian group
2. ${\displaystyle v(a+b)=va+vb}$  for ${\displaystyle v\in V}$  and ${\displaystyle a,b\in F}$
3. ${\displaystyle a(v+u)=av+au}$  for ${\displaystyle v,u\in V}$  and ${\displaystyle a\in F}$
4. ${\displaystyle (ab)v=a(bv)}$
5. ${\displaystyle 1_{F}v=v}$

The scalar multiplication is formally defined by ${\displaystyle F\times V{\xrightarrow {\phi }}V}$ , where ${\displaystyle \phi ((f,v))=fv\in V}$ .

Elements in F are called scalars, while elements in V are called vectors.

Some Properties of Vector Spaces
1. ${\displaystyle 0_{F}v=0_{V}=a0_{V}}$
2. ${\displaystyle (-1_{F})v=-v}$
3. ${\displaystyle av=0\iff a=0{\text{ or }}v=0}$
Proofs:
1. ${\displaystyle 0_{F}v=0_{F}v+0_{V}=(0_{F}+0_{F})v=0_{F}v+0_{F}v\Rightarrow 0_{V}=0_{F}v.Also,a0_{V}=a0_{V}+0_{V}=a(0_{V}+0_{V})=a0_{V}v+a0_{V}\Rightarrow a0_{V}=0_{V}v}$
2. We want to show that ${\displaystyle v+(-1_{F})v=0_{V}}$ , but ${\displaystyle v+(-1_{F})v=1_{F}v+(-1_{F})v=(1_{F}+(-1_{F}))v=0_{F}v=0_{V}}$
3. Suppose ${\displaystyle av=0}$  such that ${\displaystyle a\neq 0}$ , then ${\displaystyle a^{-1}(av)=a^{-1}0=0\Rightarrow 1_{F}v=v=0}$