Vector space: Difference between revisions
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Created page with "== Mathematical definition == Set $\mathbf{V}$ is a '''vector space''' over $\mathbb{F}$ if: 1. Are defined operations ** of addition: $\mathbf{V} \times \mathbf{V} \to \mathbf{V}$ ** of scalar multiplication: $\mathbb{F} \times \mathbf{V} \to \mathbf{V}$ 2. Are satisfied the properties ** Commutativity: $\forall u,v\in\mathbf{V},\; u+v=v+u$; ** Associativity: $\forall u,v,w\in\mathbf{V},\;\forall a,b\in\mathbb{F},\; (u+v)+w=u+(v+w)\;\land\;(ab)v=a(bv)$; ** Additive i..." |
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** Multiplicative identity: $\forall v\in\mathbf{V},\; 1v=v$; | ** Multiplicative identity: $\forall v\in\mathbf{V},\; 1v=v$; | ||
** Distributivity: $\forall u,v\in\mathbf{V},\;\forall a,b\in\mathbb{F},\; a(u+v)=au+av\;\land\;(a+b)u=au+bu$. | ** Distributivity: $\forall u,v\in\mathbf{V},\;\forall a,b\in\mathbb{F},\; a(u+v)=au+av\;\land\;(a+b)u=au+bu$. | ||
<ref>{{#cite:Q1866}}</ref> | |||
Revision as of 10:32, 29 September 2026
Mathematical definition
Set $\mathbf{V}$ is a vector space over $\mathbb{F}$ if:
1. Are defined operations
- of addition: $\mathbf{V} \times \mathbf{V} \to \mathbf{V}$
- of scalar multiplication: $\mathbb{F} \times \mathbf{V} \to \mathbf{V}$
2. Are satisfied the properties
- Commutativity: $\forall u,v\in\mathbf{V},\; u+v=v+u$;
- Associativity: $\forall u,v,w\in\mathbf{V},\;\forall a,b\in\mathbb{F},\; (u+v)+w=u+(v+w)\;\land\;(ab)v=a(bv)$;
- Additive identity: $\exists \vec 0\in\mathbf{V},\;\forall v\in\mathbf{V},\; \vec 0+v=v$;
- Additive inverse: $\forall v\in\mathbf{V},\;\exists w\in\mathbf{V},\; v+w=0$;
- Multiplicative identity: $\forall v\in\mathbf{V},\; 1v=v$;
- Distributivity: $\forall u,v\in\mathbf{V},\;\forall a,b\in\mathbb{F},\; a(u+v)=au+av\;\land\;(a+b)u=au+bu$.