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#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Thomas Breuer.
##
## Copyright of GAP belongs to its developers, whose names are too numerous
## to list here. Please refer to the COPYRIGHT file for details.
##
## SPDX-License-Identifier: GPL-2.0-or-later
##
## 1. Single Linear Mappings
## 2. Vector Spaces of Linear Mappings
##
#############################################################################
##
## <#GAPDoc Label="[1]{vspchom}">
## <E>Vector space homomorphisms</E> (or <E>linear mappings</E>) are defined
## in Section <Ref Sect="Linear Mappings"/>.
## &GAP; provides special functions to construct a particular linear
## mapping from images of given elements in the source,
## from a matrix of coefficients, or as a natural epimorphism.
## <P/>
## <M>F</M>-linear mappings with same source and same range can be added,
## so one can form vector spaces of linear mappings.
## <#/GAPDoc>
##
#############################################################################
##
## 1. Single Linear Mappings
##
#############################################################################
##
#O LeftModuleGeneralMappingByImages( <V>, <W>, <gens>, <imgs> )
##
## <#GAPDoc Label="LeftModuleGeneralMappingByImages">
## <ManSection>
## <Oper Name="LeftModuleGeneralMappingByImages" Arg='V, W, gens, imgs'/>
##
## <Description>
## Let <A>V</A> and <A>W</A> be two left modules over the same left acting
## domain <M>R</M> and <A>gens</A> and <A>imgs</A> lists
## (of the same length) of elements in <A>V</A> and <A>W</A>, respectively.
## <Ref Oper="LeftModuleGeneralMappingByImages"/> returns
## the general mapping with source <A>V</A> and range <A>W</A>
## that is defined by mapping the elements in <A>gens</A> to the
## corresponding elements in <A>imgs</A>,
## and taking the <M>R</M>-linear closure.
## <P/>
## <A>gens</A> need not generate <A>V</A> as a left <M>R</M>-module,
## and if the specification does not define a linear mapping then the result
## will be multi-valued; hence in general it is not a mapping
## (see <Ref Filt="IsMapping"/>).
## <Example><![CDATA[
## gap> V:= Rationals^2;;
## gap> W:= VectorSpace( Rationals, [ [1,2,3], [1,0,1] ] );;
## gap> f:= LeftModuleGeneralMappingByImages( V, W,
## > [[1,0],[2,0]], [[1,0,1],[1,0,1] ] );
## [ [ 1, 0 ], [ 2, 0 ] ] -> [ [ 1, 0, 1 ], [ 1, 0, 1 ] ]
## gap> IsMapping( f );
## false
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "LeftModuleGeneralMappingByImages",
[ IsLeftModule, IsLeftModule, IsHomogeneousList, IsHomogeneousList ] );
#############################################################################
##
#F LeftModuleHomomorphismByImages( <V>, <W>, <gens>, <imgs> )
#O LeftModuleHomomorphismByImagesNC( <V>, <W>, <gens>, <imgs> )
##
## <#GAPDoc Label="LeftModuleHomomorphismByImages">
## <ManSection>
## <Func Name="LeftModuleHomomorphismByImages" Arg='V, W, gens, imgs'/>
## <Oper Name="LeftModuleHomomorphismByImagesNC" Arg='V, W, gens, imgs'/>
##
## <Description>
## Let <A>V</A> and <A>W</A> be two left modules over the same left acting
## domain <M>R</M> and <A>gens</A> and <A>imgs</A> lists (of the same
## length) of elements in <A>V</A> and <A>W</A>, respectively.
## <Ref Func="LeftModuleHomomorphismByImages"/> returns
## the left <M>R</M>-module homomorphism with source <A>V</A> and range
## <A>W</A> that is defined by mapping the elements in <A>gens</A> to the
## corresponding elements in <A>imgs</A>.
## <P/>
## If <A>gens</A> does not generate <A>V</A> or if the homomorphism does not
## exist (i.e., if mapping the generators describes only a multi-valued
## mapping) then <K>fail</K> is returned.
## For creating a possibly multi-valued mapping from <A>V</A> to <A>W</A>
## that respects addition, multiplication, and scalar multiplication,
## <Ref Oper="LeftModuleGeneralMappingByImages"/> can be used.
## <P/>
## <Ref Oper="LeftModuleHomomorphismByImagesNC"/> does the same as
## <Ref Func="LeftModuleHomomorphismByImages"/>,
## except that it omits all checks.
## <Example><![CDATA[
## gap> V:=Rationals^2;;
## gap> W:=VectorSpace( Rationals, [ [ 1, 0, 1 ], [ 1, 2, 3 ] ] );;
## gap> f:=LeftModuleHomomorphismByImages( V, W,
## > [ [ 1, 0 ], [ 0, 1 ] ], [ [ 1, 0, 1 ], [ 1, 2, 3 ] ] );
## [ [ 1, 0 ], [ 0, 1 ] ] -> [ [ 1, 0, 1 ], [ 1, 2, 3 ] ]
## gap> Image( f, [1,1] );
## [ 2, 2, 4 ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "LeftModuleHomomorphismByImages" );
DeclareOperation( "LeftModuleHomomorphismByImagesNC",
[ IsLeftModule, IsLeftModule, IsList, IsList ] );
#############################################################################
##
#A AsLeftModuleGeneralMappingByImages( <map> )
##
## <ManSection>
## <Attr Name="AsLeftModuleGeneralMappingByImages" Arg='map'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareAttribute( "AsLeftModuleGeneralMappingByImages", IsGeneralMapping );
#############################################################################
##
#O LeftModuleHomomorphismByMatrix( <BS>, <matrix>, <BR> )
##
## <#GAPDoc Label="LeftModuleHomomorphismByMatrix">
## <ManSection>
## <Oper Name="LeftModuleHomomorphismByMatrix" Arg='BS, matrix, BR'/>
##
## <Description>
## Let <A>BS</A> and <A>BR</A> be bases of the left <M>R</M>-modules
## <M>V</M> and <M>W</M>, respectively.
## <Ref Oper="LeftModuleHomomorphismByMatrix"/> returns the <M>R</M>-linear
## mapping from <M>V</M> to <M>W</M> that is defined by the matrix
## <A>matrix</A>, as follows.
## The image of the <M>i</M>-th basis vector of <A>BS</A> is the linear
## combination of the basis vectors of <A>BR</A> with coefficients the
## <M>i</M>-th row of <A>matrix</A>.
## <Example><![CDATA[
## gap> V:= Rationals^2;;
## gap> W:= VectorSpace( Rationals, [ [ 1, 0, 1 ], [ 1, 2, 3 ] ] );;
## gap> f:= LeftModuleHomomorphismByMatrix( Basis( V ),
## > [ [ 1, 2 ], [ 3, 1 ] ], Basis( W ) );
## <linear mapping by matrix, ( Rationals^
## 2 ) -> <vector space over Rationals, with 2 generators>>
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "LeftModuleHomomorphismByMatrix",
[ IsBasis, IsMatrix, IsBasis ] );
#############################################################################
##
#O NaturalHomomorphismBySubspace( <V>, <W> ) . . . . . map onto factor space
##
## <#GAPDoc Label="NaturalHomomorphismBySubspace">
## <ManSection>
## <Oper Name="NaturalHomomorphismBySubspace" Arg='V, W'/>
##
## <Description>
## For an <M>R</M>-vector space <A>V</A> and a subspace <A>W</A> of
## <A>V</A>,
## <Ref Oper="NaturalHomomorphismBySubspace"/> returns the <M>R</M>-linear
## mapping that is the natural projection of <A>V</A> onto the factor space
## <C><A>V</A> / <A>W</A></C>.
## <Example><![CDATA[
## gap> V:= Rationals^3;;
## gap> W:= VectorSpace( Rationals, [ [ 1, 1, 1 ] ] );;
## gap> f:= NaturalHomomorphismBySubspace( V, W );
## <linear mapping by matrix, ( Rationals^3 ) -> ( Rationals^2 )>
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "NaturalHomomorphismBySubspace",
[ IsLeftModule, IsLeftModule ] );
#############################################################################
##
#F NaturalHomomorphismBySubspaceOntoFullRowSpace( <V>, <W> )
##
## <ManSection>
## <Func Name="NaturalHomomorphismBySubspaceOntoFullRowSpace" Arg='V, W'/>
##
## <Description>
## returns a vector space homomorphism from the vector space <A>V</A> onto a
## full row space, with kernel exactly the vector space <A>W</A>,
## which must be contained in <A>V</A>.
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "NaturalHomomorphismBySubspaceOntoFullRowSpace" );
#############################################################################
##
## 2. Vector Spaces of Linear Mappings
##
#############################################################################
##
#P IsFullHomModule( <M> )
##
## <#GAPDoc Label="IsFullHomModule">
## <ManSection>
## <Prop Name="IsFullHomModule" Arg='M'/>
##
## <Description>
## A <E>full hom module</E> is a module of all <M>R</M>-linear mappings
## between two left <M>R</M>-modules.
## The function <Ref Oper="Hom"/> can be used to construct a full hom
## module.
## <Example><![CDATA[
## gap> V:= Rationals^2;;
## gap> W:= VectorSpace( Rationals, [ [ 1, 0, 1 ], [ 1, 2, 3 ] ] );;
## gap> H:= Hom( Rationals, V, W );;
## gap> IsFullHomModule( H );
## true
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsFullHomModule", IsFreeLeftModule );
#############################################################################
##
#P IsPseudoCanonicalBasisFullHomModule( <B> )
##
## <#GAPDoc Label="IsPseudoCanonicalBasisFullHomModule">
## <ManSection>
## <Prop Name="IsPseudoCanonicalBasisFullHomModule" Arg='B'/>
##
## <Description>
## A basis of a full hom module is called pseudo canonical basis
## if the matrices of its basis vectors w.r.t. the stored bases of source
## and range contain exactly one identity entry and otherwise zeros.
## <P/>
## Note that this is not a canonical basis
## (see <Ref Attr="CanonicalBasis"/>)
## because it depends on the stored bases of source and range.
## <Example><![CDATA[
## gap> IsPseudoCanonicalBasisFullHomModule( Basis( H ) );
## true
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsPseudoCanonicalBasisFullHomModule", IsBasis );
#############################################################################
##
#O Hom( <F>, <V>, <W> ) . . . space of <F>-linear mappings from <V> to <W>
##
## <#GAPDoc Label="Hom">
## <ManSection>
## <Oper Name="Hom" Arg='F, V, W'/>
##
## <Description>
## For a field <A>F</A> and two vector spaces <A>V</A> and <A>W</A>
## that can be regarded as <A>F</A>-modules
## (see <Ref Oper="AsLeftModule"/>),
## <Ref Oper="Hom"/> returns the <A>F</A>-vector space of
## all <A>F</A>-linear mappings from <A>V</A> to <A>W</A>.
## <Example><![CDATA[
## gap> V:= Rationals^2;;
## gap> W:= VectorSpace( Rationals, [ [ 1, 0, 1 ], [ 1, 2, 3 ] ] );;
## gap> H:= Hom( Rationals, V, W );
## Hom( Rationals, ( Rationals^2 ), <vector space over Rationals, with
## 2 generators> )
## gap> Dimension( H );
## 4
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "Hom", [ IsRing, IsLeftModule, IsLeftModule ] );
#############################################################################
##
#O End( <F>, <V> ) . . . . . . space of <F>-linear mappings from <V> to <V>
##
## <#GAPDoc Label="End">
## <ManSection>
## <Oper Name="End" Arg='F, V'/>
##
## <Description>
## For a field <A>F</A> and a vector space <A>V</A> that can be regarded as
## an <A>F</A>-module (see <Ref Oper="AsLeftModule"/>),
## <Ref Oper="End"/> returns the <A>F</A>-algebra of all <A>F</A>-linear
## mappings from <A>V</A> to <A>V</A>.
## <Example><![CDATA[
## gap> A:= End( Rationals, Rationals^2 );
## End( Rationals, ( Rationals^2 ) )
## gap> Dimension( A );
## 4
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "End", [ IsRing, IsLeftModule ] );
#############################################################################
##
#F IsLinearMappingsModule( <V> )
##
## <#GAPDoc Label="IsLinearMappingsModule">
## <ManSection>
## <Filt Name="IsLinearMappingsModule" Arg='V'/>
##
## <Description>
## If an <M>F</M>-vector space <A>V</A> is in the filter
## <Ref Filt="IsLinearMappingsModule"/> then
## this expresses that <A>V</A> consists of linear mappings,
## and that <A>V</A> is handled via the mechanism of nice bases
## (see <Ref Sect="Vector Spaces Handled By Nice Bases"/>),
## in the following way.
## Let <M>S</M> and <M>R</M> be the source and the range, respectively,
## of each mapping in <M>V</M>.
## Then the <Ref Attr="NiceFreeLeftModuleInfo"/> value of <A>V</A> is
## a record with the components <C>basissource</C> (a basis <M>B_S</M> of
## <M>S</M>) and <C>basisrange</C> (a basis <M>B_R</M> of <M>R</M>),
## and the <Ref Oper="NiceVector"/> value of <M>v \in <A>V</A></M>
## is defined as the matrix of the <M>F</M>-linear mapping <M>v</M>
## w.r.t. the bases <M>B_S</M> and <M>B_R</M>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareHandlingByNiceBasis( "IsLinearMappingsModule",
"for free left modules of linear mappings" );
#############################################################################
##
#M IsFiniteDimensional( <A> ) . . . . . hom FLMLORs are finite dimensional
##
InstallTrueMethod( IsFiniteDimensional, IsLinearMappingsModule );
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