vector space
Table of Contents
1. Introduction
A vector space \(V\) is a set with addition and scalar multiplication defined. It obeys the following axioms:
\begin{align} \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{align}vector spaces are an Abelian group under addition. \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) are considered vectors so long as they fulfill these properties.