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Vector space

Definition

A vector space $V$ over a field $\mathbb{F}$ is an abelian group acted on by the field by "scalar multiplication" $\mathbb{F} \times V \to V$, where $\forall u,v \in V; a,b \in \mathbb{F}$:

• $a(u+v)=au+av$
• $(a+b)v=av+bv$
• $a(bv)=(ab)v$
• $1v=v$

Elements of $V$ are called vectors and elements of $\mathbb{F}$ are called scalars.

In other words, a vector space is a module over a field.

Criterion

The first two criteria can be condensed into $a(u+v)=au+av$.

Examples

Given a field $\mathbb{F}$, the canonical vector space $\mathbb{F}^n$ for some $n \in \mathbb{N}$.

If $F \subset E$ are fields, then $E$ can be considered as a vector space over $F$.

all these pages adapted with probably insufficient credit from my university's lecture notes