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#############################################################################
##
#W vspchom.gd GAP library Thomas Breuer
##
#H @(#)$Id: vspchom.gd,v 4.17 2002/04/15 10:05:29 sal Exp $
##
#Y Copyright (C) 1997, Lehrstuhl D fuer Mathematik, RWTH Aachen, Germany
#Y (C) 1998 School Math and Comp. Sci., University of St. Andrews, Scotland
#Y Copyright (C) 2002 The GAP Group
##
## 1. Single Linear Mappings
## 2. Vector Spaces of Linear Mappings
##
Revision.vspchom_gd :=
"@(#)$Id: vspchom.gd,v 4.17 2002/04/15 10:05:29 sal Exp $";
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#1
## *Vector space homomorphisms* (or *linear mappings*) are defined in
## Section~"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.
##
## $F$-linear mappings with same source and same range can be added,
## so one can form vector spaces of linear mappings.
##
#############################################################################
##
## 1. Single Linear Mappings
##
#############################################################################
##
#O LeftModuleGeneralMappingByImages( <V>, <W>, <gens>, <imgs> )
##
## Let <V> and <W> be two left modules over the same left acting domain
## $R$, say, and <gens> and <imgs> lists (of the same length)
## of elements in <V> and <W>, respectively.
## `LeftModuleGeneralMappingByImages' returns the general mapping
## with source <V> and range <W> that is defined by mapping the elements in
## <gens> to the corresponding elements in <imgs>,
## and taking the $R$-linear closure.
##
## <gens> need not generate <V> as a left $R$-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~"IsMapping").
##
DeclareOperation( "LeftModuleGeneralMappingByImages",
[ IsLeftModule, IsLeftModule, IsHomogeneousList, IsHomogeneousList ] );
#############################################################################
##
#F LeftModuleHomomorphismByImages( <V>, <W>, <gens>, <imgs> )
#O LeftModuleHomomorphismByImagesNC( <V>, <W>, <gens>, <imgs> )
##
## Let <V> and <W> be two left modules over the same left acting domain
## $R$, say, and <gens> and <imgs> lists (of the same length)
## of elements in <V> and <W>, respectively.
## `LeftModuleHomomorphismByImages' returns the left $R$-module homomorphism
## with source <V> and range <W> that is defined by mapping the elements in
## <gens> to the corresponding elements in <imgs>.
##
## If <gens> does not generate <V> or if the homomorphism does not exist
## (i.e., if mapping the generators describes only a multi-valued mapping)
## then `fail' is returned.
## For creating a possibly multi-valued mapping from <V> to <W> that
## respects addition, multiplication, and scalar multiplication,
## `LeftModuleGeneralMappingByImages' can be used.
##
## `LeftModuleHomomorphismByImagesNC' does the same as
## `LeftModuleHomomorphismByImages', except that it omits all checks.
##
DeclareGlobalFunction( "LeftModuleHomomorphismByImages" );
DeclareOperation( "LeftModuleHomomorphismByImagesNC",
[ IsLeftModule, IsLeftModule, IsList, IsList ] );
#############################################################################
##
#A AsLeftModuleGeneralMappingByImages( <map> )
##
DeclareAttribute( "AsLeftModuleGeneralMappingByImages", IsGeneralMapping );
#############################################################################
##
#O LeftModuleHomomorphismByMatrix( <BS>, <matrix>, <BR> )
##
## Let <BS> and <BR> be bases of the left $R$-modules $V$ and $W$,
## respectively.
## `LeftModuleHomomorphismByMatrix' returns the $R$-linear mapping from $V$
## to $W$ that is defined by the matrix <matrix> as follows.
## The image of the $i$-th basis vector of <BS> is the linear combination of
## the basis vectors of <BR> with coefficients the $i$-th row of <matrix>.
##
DeclareOperation( "LeftModuleHomomorphismByMatrix",
[ IsBasis, IsMatrix, IsBasis ] );
#############################################################################
##
#O NaturalHomomorphismBySubspace( <V>, <W> ) . . . . . map onto factor space
##
## For an $R$-vector space <V> and a subspace <W> of <V>,
## `NaturalHomomorphismBySubspace' returns the $R$-linear mapping that is
## the natural projection of <V> onto the factor space `<V> / <W>'.
##
DeclareOperation( "NaturalHomomorphismBySubspace",
[ IsLeftModule, IsLeftModule ] );
#############################################################################
##
#F NaturalHomomorphismBySubspaceOntoFullRowSpace( <V>, <W> )
##
## returns a vector space homomorphism from the vector space <V> onto a full
## row space, with kernel exactly the vector space <W>,
## which must be contained in <V>.
##
DeclareGlobalFunction( "NaturalHomomorphismBySubspaceOntoFullRowSpace" );
#############################################################################
##
## 2. Vector Spaces of Linear Mappings
##
#############################################################################
##
#P IsFullHomModule( <M> )
##
## A *full hom module* is a module of all $R$-linear mappings between two
## left $R$-modules. The function `Hom' (see~"Hom") can be used to
## construct a full hom module.
##
DeclareProperty( "IsFullHomModule", IsFreeLeftModule );
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##
#P IsPseudoCanonicalBasisFullHomModule( <B> )
##
## 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.
##
## Note that this is not a canonical basis (see~"CanonicalBasis")
## because it depends on the stored bases of source and range.
##
DeclareProperty( "IsPseudoCanonicalBasisFullHomModule", IsBasis );
#############################################################################
##
#O Hom( <F>, <V>, <W> ) . . . space of <F>-linear mappings from <V> to <W>
##
## For a field <F> and two vector spaces <V> and <W> that can be regarded as
## <F>-modules (see~"AsLeftModule"), `Hom' returns the <F>-vector space of
## all <F>-linear mappings from <V> to <W>.
##
DeclareOperation( "Hom", [ IsRing, IsLeftModule, IsLeftModule ] );
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##
#O End( <F>, <V> ) . . . . . . space of <F>-linear mappings from <V> to <V>
##
## For a field <F> and a vector space <V> that can be regarded as an
## <F>-module (see~"AsLeftModule"), `End' returns the <F>-algebra of
## all <F>-linear mappings from <V> to <V>.
##
DeclareOperation( "End", [ IsRing, IsLeftModule ] );
#############################################################################
##
#F IsLinearMappingsModule( <V> )
##
## If an $F$-vector space <V> is in the filter `IsLinearMappingsModule' then
## this expresses that <V> consists of linear mappings, and that <V> is
## handled via the mechanism of nice bases
## (see~"Vector Spaces Handled By Nice Bases") in the following way.
## Let $S$ and $R$ be the source and the range, respectively, of each
## mapping in $V$.
## Then the `NiceFreeLeftModuleInfo' value of <V> is a record with the
## components `basissource' (a basis $B_S$ of $S$)
## and `basisrange' (a basis $B_R$ of $R$),
## and the `NiceVector' value of $v \in <V>$ is defined as the
## matrix of the $F$-linear mapping $v$ w.r.t.~the bases $B_S$ and $B_R$.
##
DeclareHandlingByNiceBasis( "IsLinearMappingsModule",
"for free left modules of linear mappings" );
#############################################################################
##
#M IsFiniteDimensional( <A> ) . . . . . hom FLMLORs are finite dimensional
##
InstallTrueMethod( IsFiniteDimensional, IsLinearMappingsModule );
#############################################################################
##
#E
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