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Vector Space

A vector space $V$ over field $F$ is an abelian group $V$ with an operation called scalar multiplication defined on field $F$.

Scalar multiplication is as defined as $V \times F \rightarrow F$ and it should satisfy the following axioms

  1. $a(b\vec{v}) = (ab)\vec{v}$
  2. $1\vec{v}=\vec{v}$
  3. $(a+b)\vec{v} =a\vec{v}+b\vec{v}$
  4. $a(\vec{u}+\vec{v})=a\vec{u}+a\vec{v}$

These are natural derivations such that the field and group properties remain intact.