Axioms (Vector Space)

To qualify as a vector space, a set V and its associated operations of addition (+) and multiplication/scaling (⋅) must adhere to the below:

Associativity #

u+(v+w)=(u+v)+w

Commutivity #

u+v=v+u

Identity of Addition #

There exists and element 0∈V, called the zero vector, such that v+0=v for all v∈V.

Inverse of Addition #

For every v∈V, there exists an element −v∈V, such that v+(−v)=0.

Compatibility #

a(bv)=(ab)v

Scalar/Multiplication Identify #

1v=v

where 1 denotes the scalar identity

Distributivity #

a(u+v)=au+av