Abstract Algebra/Vector Spaces

Definition (Vector Space)

Let F be a field. A set V with two binary operations: + (addition) and $\times$ (scalar multiplication), is called a Vector Space if it has the following properties:

1. $(V,+)$ forms an abelian group
2. $(a+b)v=av+bv$ for $a,b\in F$ and $v\in V$
3. $a(v+u)=av+au$ for $a\in F$ and $v,u\in V$
4. $(ab)v=a(bv)$
5. $1_{F}v=v$

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

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

Some Properties of Vector Spaces

1. $0_{F}v=0_{V}=a0_{V}$
2. $(-1_{F})v=-v$
3. $av=0\iff a=0{\text{ or }}v=0$

Proofs:

1. $0_{F}v=(0_{F}+0_{F})v=0_{F}v+0_{F}v\Rightarrow 0_{V}=0_{F}v.Also,a0_{V}=a(0_{V}+0_{V})=a0_{V}+a0_{V}\Rightarrow a0_{V}=0_{V}$
2. We want to show that $v+(-1_{F})v=0_{V}$, but $v+(-1_{F})v=1_{F}v+(-1_{F})v=(1_{F}+(-1_{F}))v=0_{F}v=0_{V}$
3. Suppose $av=0$ such that $a\neq 0$, then $a^{-1}(av)=a^{-1}0=0\Rightarrow 1_{F}v=v=0$