Vector space: Difference between revisions

Revision as of 10:51, 29 September 2026

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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

@@ Line 1: / Line 1: @@
-== Overview ==
+== Mathematical definition ==
-mathematical concept
+Set $\mathbf{V}$ is a '''vector space''' over $\mathbb{F}$ if:
+. Are defined operations
+** of addition: $\mathbf{V} \times \mathbf{V} \to \mathbf{V}$
+** of scalar multiplication: $\mathbb{F} \times \mathbf{V} \to \mathbf{V}$
+. 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: $\forall u,v\in\mathbf{V},\;\forall a,b\in\mathbb{F},\; a(u+v)=au+av\;\land\;(a+b)u=au+bu$.
+<ref>{{#cite:Q1866}}</ref>