:PROPERTIES:
:ID:       ab024db7-6903-48ee-98fc-b2a228709c04
:ROAM_ALIASES: vector linear "linear space"
:END:
#+title: vector space
#+author: Preston Pan
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* 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 [[id:ba7b95b0-0ce6-4b33-9a79-5e5fddaea710][group]] under addition. $\vec{a}$, $\vec{b}$, and $\vec{c}$ are considered vectors so long as they fulfill these
properties.