# 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 $a^{-1}(av)=a^{-1}0=0\Rightarrow 1_{F}v=v=0$ 