Vector Space Axioms

Axioms

A Vector Space is defined as a set V\mathbb{V}, together with two operations:

  1. Vector addition: +:V×V→V+: \mathbb{V} \times \mathbb{V} \to \mathbb{V}
  2. Scalar multiplication: ⋅:F×V→V\cdot: \mathbb{F} \times \mathbb{V} \to \mathbb{V}

where F\mathbb{F} is a field (typically R\mathbb{R} or C\mathbb{C}).

For arbitrary vectors u,v,w∈Vu,v,w \in \mathbb{V} and scalars a,b∈Fa,b \in \mathbb{F}, the following axioms must hold:

Vector Addition Axioms

  1. Associativity
(u+v)+w=u+(v+w)(u + v) + w = u + (v + w)
  1. Additive Identity
    • There exists a zero vector 0∈V\mathbf{0} \in \mathbb{V} such that:
v+0=0+v=v for all v∈Vv + \mathbf{0} = \mathbf{0} + v = v \text{ for all } v \in \mathbb{V}
  1. Additive Inverse
    • For each v∈Vv \in \mathbb{V}, there exists an element −v∈V-v \in \mathbb{V} such that:
v+(−v)=(−v)+v=0v + (-v) = (-v) + v = \mathbf{0}
  1. Commutativity
u+v=v+uu + v = v + u

Scalar Multiplication Axioms

  1. Scalar Multiplication Identity
1v=v for all v∈V1v = v \text{ for all } v \in \mathbb{V}
  1. Scalar Multiplication Associativity
a(bv)=(ab)va(bv) = (ab)v
  1. Distribution over Vector Addition
a(u+v)=au+ava(u + v) = au + av
  1. Distribution over Scalar Addition
(a+b)v=av+bv(a + b)v = av + bv