# 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.

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

### Constructors

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

Return the free module over the ring $R$ with the given rank.

VectorSpaceMethod
VectorSpace(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 = 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)