Abstract Algebra/Vector Spaces

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. ${\displaystyle (a+b)v=av+bv}$ for ${\displaystyle a,b\in F}$ and ${\displaystyle v\in V}$
3. ${\displaystyle a(v+u)=av+au}$ for ${\displaystyle a\in F}$ and ${\displaystyle v,u\in V}$
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. ${\displaystyle 0_{F}v=(0_{F}+0_{F})v=0_{F}v+0_{F}v\Rightarrow 0_{V}=0_{F}v.Also,a0_{V}=a(0_{V}+0_{V})=a0_{V}+a0_{V}\Rightarrow a0_{V}=0_{V}}$
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 ${\displaystyle a^{-1}(av)=a^{-1}0=0\Rightarrow 1_{F}v=v=0}$