Free Modules and Vector Spaces

AbstractAlgebra allows the construction of the free module of any rank over any Euclidean ring and the vector space of any dimension over a field. By default the system considers the free module of a given rank over a given ring or vector space of given dimension over a field to be unique.

Types and parents

AbstractAlgebra provides the type FreeModule{T} for free modules, where T is the type of the elements of the ring $R$ over which the module is built. The type FreeModule{T} belongs to AbstractAlgebra.FPModule{T}.

Vector spaces are simply free modules over a field.

The free module of a given rank over a given ring is made unique on the system by caching them (unless an optional cache parameter is set to false).

See src/generic/GenericTypes.jl for an example of how to implement such a cache (which usually makes use of a dictionary).

Functionality for free modules

As well as implementing the entire module interface, free modules provide the following functionality.

Constructors

FreeModule(R::NCRing, rank::Int; cached::Bool = true)

VectorSpace(R::Field, dim::Int; cached::Bool = true)

Construct the free module/vector space of given rank/dimension.

Examples

julia> M = FreeModule(ZZ, 3)
Free module of rank 3 over Integers

julia> V = VectorSpace(QQ, 2)
Vector space of dimension 2 over Rationals

Basic manipulation

rank(M::Generic.FreeModule{T}) where T <: AbstractAlgebra.RingElem
dim(V::Generic.FreeModule{T}) where T <: AbstractAlgebra.FieldElem
basis(V::Generic.FreeModule{T}) where T <: AbstractAlgebra.FieldElem

Examples

julia> M = FreeModule(ZZ, 3)
Free module of rank 3 over Integers

julia> V = VectorSpace(QQ, 2)
Vector space of dimension 2 over Rationals

julia> rank(M)
3

julia> dim(V)
2

julia> basis(V)
2-element Array{AbstractAlgebra.Generic.FreeModuleElem{Rational{BigInt}},1}:
 (1//1, 0//1)
 (0//1, 1//1)