Vector Space
Definition
Suppose that
- AC Additive Closure
Ifu,v∈V , thenu+v∈V . - SC Scalar Closure
Ifα∈C andu∈V , thenαu∈V . - C Commutativity
Ifu,v∈V , thenu+v=v+u . - AA Additive Associativity
Ifu,v,w∈V , thenu+(v+w)=(u+v)+w . - Z Zero Vector
There is a vector,0 , called the zero vector, such thatu+0=u for allu∈V . - AI Additive Inverses
Ifu∈V , then there exists a vector−u∈V so thatu+(−u)=0 . - SMA Scalar Multiplication Associativity
Ifα,β∈C andu∈V , thenα(βu)=(αβ)u . - DVA Distributivity across Vector Addition
Ifα∈C andu,v∈V , thenα(u+v)=αu+αv . - DSA Distributivity across Scalar Addition
Ifα,β∈C andu∈V , then(α+β)u=αu+βu . - O One
Ifu∈V , then1u=u .
The objects in V are called vectors, no matter what else they might really be, simply by virtue of being elements of a vector space.
