Free Modules and Vector Spaces

AbstractAlgebra allows the construction of free modules 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.

Generic free module and vector space types

AbstractAlgebra provides generic types for free modules and vector spaces, via 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.

Elements of a free module have type FreeModuleElem{T}.

Vector spaces are simply free modules over a field.

The implementation of generic free modules can be found in src/generic/FreeModule.jl.

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).

Abstract types

The type FreeModule{T} belongs to FPModule{T} and FreeModuleElem{T} to FPModuleElem{T}. Here the FP prefix stands for finitely presented.

Functionality for free modules

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

Constructors

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 <: RingElem
dim(V::Generic.FreeModule{T}) where T <: FieldElem
basis(V::Generic.FreeModule{T}) where T <: 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 Vector{AbstractAlgebra.Generic.FreeModuleElem{Rational{BigInt}}}:
 (1//1, 0//1)
 (0//1, 1//1)