Abstract algebra/Vector Spaces

Definition

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

The scalar multiplication is formerly defined by $F\times V\stackrel{\varphi }{\to }V$, where $\varphi ((f,v))=fv\in V$.

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

Some Properties of Vector Spaces

1. $0_{F}v=0_V=a{0}_V$
2. $(-1_F)v=-v$
3. $av=0⟺a=0$ or $v=0$

Proofs:

1. $0_{F}v=0_{F}v+0_V=(0_F+0_F)v=0_{F}v+0_{F}v\Rightarrow 0_V=0_{F}v.Also,a{0}_V=a{0}_V+0_V=a(0_V+0_V)=a{0}_{V}v+a{0}_V\Rightarrow a{0}_V=0_{V}v$
2. We want to show that $v+(-1_F)v=0$, 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 $a\ne 0$, then $a^{-1}(av)=a^{-1}0=0\Rightarrow 1_{F}v=v=0$