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Vector space

Revision as of 13:05, 4 November 2006 by Bubka (talk | contribs) (→‎Axioms of a vector space)

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A vector space over a field (frequently the real numbers) is an object which arises in linear algebra and abstract algebra. A vector space $V$ over a field $F$ consists of a set (of vectors) and two operations, vector addition and scalar multiplication, which obey the following rules:

Axioms of a vector space

  • Scalar multiplication is associative, so if $r, s \in F$ and ${\mathbf v} \in V$ then $(rs){\mathbf v} = r(s{\mathbf v})$.
  • Scalar multiplication is distributive over both vector and scalar addition, so if $r \in F$ and ${\mathbf v, w} \in V$ then $r({\mathbf v + w}) = r{\mathbf v} + r{\mathbf w}$.
  • if $x \in V$, $1.{\mathbf x}={\mathbf x}$

Examples of vector spaces

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