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

Under vector addition, the set of vectors forms an abelian group. Thus, addition is associative and commutative and there is an additive identity (usually denoted $\mathbf 0$ ) and additive inverses.

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 $x \in V$ , $1.{\mathbf x}={\mathbf x}$

Subspaces

If $S \subseteq V$ , and $\mathbf S$ is a vector space itself, then it is called a subspace of $\mathbf V$ .

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Vector space

Axioms of a vector space

Subspaces