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| == Mathematical definition == | | == Overview == |
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| Set $\mathbf{V}$ is a '''vector space''' over $\mathbb{F}$ if:
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| 1. Are defined operations
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| ** of addition: $\mathbf{V} \times \mathbf{V} \to \mathbf{V}$
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| ** of scalar multiplication: $\mathbb{F} \times \mathbf{V} \to \mathbf{V}$
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| 2. Are satisfied the properties
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| ** Commutativity: $\forall u,v\in\mathbf{V},\; u+v=v+u$;
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| ** Associativity: $\forall u,v,w\in\mathbf{V},\;\forall a,b\in\mathbb{F},\; (u+v)+w=u+(v+w)\;\land\;(ab)v=a(bv)$;
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| ** Additive identity: $\exists \vec 0\in\mathbf{V},\;\forall v\in\mathbf{V},\; \vec 0+v=v$;
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| ** Additive inverse: $\forall v\in\mathbf{V},\;\exists w\in\mathbf{V},\; v+w=0$;
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| ** Multiplicative identity: $\forall v\in\mathbf{V},\; 1v=v$;
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| ** Distributivity: $\forall u,v\in\mathbf{V},\;\forall a,b\in\mathbb{F},\; a(u+v)=au+av\;\land\;(a+b)u=au+bu$.
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| <ref>{{#cite:Q1866}}</ref>
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