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vector space

Table of Contents

1. Introduction

A vector space VV is a set with addition and scalar multiplication defined. It obeys the following axioms:

a→+(b→+c→)=(a→+b→)+c→a→+b→=b→+a→∃0→,∀a→,a→+0→=a→∀a→,∃−a→,a→+−a→=0→(cd)a→=c(da→)1a→=a→c(a→+b→)=ca→+cb→(c+d)a→=ca→+da→\begin{aligned}\label{}\vec{a} + (\vec{b} + \vec{c}) = (\vec{a} + \vec{b}) + \vec{c} \\\vec{a} + \vec{b} = \vec{b} + \vec{a} \\\exists \vec{0},\forall \vec{a}, \vec{a} + \vec{0} = \vec{a} \\\forall \vec{a},\exists\vec{-a}, \vec{a} + \vec{-a} = \vec{0} \\(cd)\vec{a} = c(d\vec{a}) \\1\vec{a} = \vec{a} \\c(\vec{a} + \vec{b}) = c\vec{a} + c\vec{b} \\(c + d)\vec{a} = c\vec{a} + d\vec{a}\end{aligned}

vector spaces are an Abelian group under addition. a→\vec{a}, b→\vec{b}, and c→\vec{c} are considered vectors so long as they fulfill these properties.