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Vector space

A set with addition and scalar multiplication satisfying the vector space axioms.
Vector space

A vector space over a F\mathbb{F} is a VV equipped with two operations ( ) +:V×V→V+:V\times V\to V and ⋅:F×V→V\cdot:\mathbb{F}\times V\to V such that for all u,v,w∈Vu,v,w\in V and a,b∈Fa,b\in\mathbb{F}:

u+v=v+u,(u+v)+w=u+(v+w),∃ 0∈V: v+0=v,∀v∈V ∃ (−v)∈V: v+(−v)=0,a⋅(u+v)=a⋅u+a⋅v,(a+b)⋅v=a⋅v+b⋅v,(ab)⋅v=a⋅(b⋅v),1⋅v=v. \begin{aligned} &u+v=v+u,\qquad (u+v)+w=u+(v+w),\\ &\exists\,0\in V:\ v+0=v,\qquad \forall v\in V\ \exists\,(-v)\in V:\ v+(-v)=0,\\ &a\cdot(u+v)=a\cdot u+a\cdot v,\qquad (a+b)\cdot v=a\cdot v+b\cdot v,\\ &(ab)\cdot v=a\cdot(b\cdot v),\qquad 1\cdot v=v. \end{aligned}

Vector spaces are the basic objects studied via and invariants such as the and of operators.

Examples:

  • Rn\mathbb{R}^n with componentwise addition and scalar multiplication is a vector space over R\mathbb{R}.
  • The set of polynomials F[x]\mathbb{F}[x] with the usual addition and scalar multiplication is a vector space over F\mathbb{F}.
  • The set of m×nm\times n matrices over F\mathbb{F} is a vector space over F\mathbb{F}.