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
AbstractAlgebra.free_module — Method
free_module(R::NCRing, rank::Int; cached::Bool = true)Return the free module over the ring $R$ with the given rank.
AbstractAlgebra.vector_space — Method
vector_space(R::Field, dim::Int; cached::Bool = true)Return the vector space over the field $R$ with the given dimension.
Construct the free module/vector space of given rank/dimension.
Examples
julia> M = free_module(ZZ, 3)
Free module of rank 3 over integers
julia> V = vector_space(QQ, 2)
Vector space of dimension 2 over rationals
Basic manipulation
LinearAlgebra.rank — Method
rank(M::FreeModule{T}) where T <: NCRingElementReturn the rank of the given free module.
Examples
julia> M = free_module(ZZ, 3)
Free module of rank 3 over integers
julia> rank(M)
3AbstractAlgebra.Generic.dim — Method
dim(M::FreeModule{T}) where T <: FieldElementReturn the dimension of the given vector space.
Examples
julia> V = vector_space(QQ, 2)
Vector space of dimension 2 over rationals
julia> dim(V)
2AbstractAlgebra.Generic.basis — Method
basis(M::FreeModule)Return the standard basis of the given free module, i.e. its generators.
Examples
julia> V = vector_space(QQ, 2)
Vector space of dimension 2 over rationals
julia> basis(V)
2-element Vector{AbstractAlgebra.Generic.FreeModuleElem{Rational{BigInt}}}:
(1//1, 0//1)
(0//1, 1//1)