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| author | Preston Pan <preston@nullring.xyz> | 2024-06-28 21:30:42 -0700 |
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| committer | Preston Pan <preston@nullring.xyz> | 2024-06-28 21:30:42 -0700 |
| commit | e7dd5245c35d2794f59bcf700a6a92009ec8c478 (patch) | |
| tree | 0d0e81552f0426f8b715bd5bd3bdd0856058db2c /mindmap/vector space.org | |
| parent | 01ba01763b81a838dcbac4c08243804e068495b9 (diff) | |
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| -rw-r--r-- | mindmap/vector space.org | 27 |
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diff --git a/mindmap/vector space.org b/mindmap/vector space.org new file mode 100644 index 0000000..55f37c8 --- /dev/null +++ b/mindmap/vector space.org @@ -0,0 +1,27 @@ +:PROPERTIES: +:ID: ab024db7-6903-48ee-98fc-b2a228709c04 +:ROAM_ALIASES: vector linear "linear space" +:END: +#+title: vector space +#+author: Preston Pan +#+html_head: <link rel="stylesheet" type="text/css" href="../style.css" /> +#+html_head: <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script> +#+html_head: <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> +#+options: broken-links:t + +* 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. |