Vector Space

Idea

A vector space is a generalization of ${\mathbb{R}}^n$¹, and is primarily of interest for studying linear maps.

Definition

A vector space is a algebraic structure $(F,V,\cdot ,{+}_V)$ where,

• $(F,{+}_F,{\cdot }_F)$ is some field.
• $(V,{+}_V)$ is an Abelian group.
• $\cdot :F\to V\to V$ is a scalar multiplication.

It must satisfy the following rules:

$$\begin{array}{rl}{1}_F\cdot \vec{x}=\vec{x}& \forall \vec{x}:V\\ a\cdot \vec{x}{+}_{V}b\cdot \vec{x}=(a{+}_{F}b)\cdot \vec{x}& \forall a,b:F,\vec{x}:V\\ a\cdot \vec{x}{+}_{V}a\cdot \vec{y}=a\cdot (\vec{x}{+}_V\vec{y})& \forall a:F,\vec{x},\vec{y}:V\\ a\cdot (b\cdot \vec{x})=(a{\cdot }_{F}b)\cdot \vec{x}& \forall a,b:F,\vec{x}:V\end{array}\text{\hspace{1em}(unit)(distr-F)(distr-V)(assoc)}$$

Where $F$ differs from $V$ then it is not needed to have subscripts on the operations and constants, as it can always be determined by context.

Relation to Modules

• One can embed all vector spaces as modules by forgetting the structure.
• Given a module over a field $F$,

References

pinter2010-abstract-algebra (ch28)

Footnotes

1. Real Numbers ↩