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#@local a,b,bf,bm,bt,bv,bw,comp1,comp2,comp3,endo,endoendo,f,funs,hom,id,m
#@local map,map1,map2,map3,map4,map5,map6,map7,map8,mb,nathom,sub,sum,t
#@local triv,v,w,zero
gap> START_TEST("vspchom.tst");
#############################################################################
##
## tests for linear mappings given by images
##
gap> f:= GF(3);
GF(3)
gap> v:= GF(27);
GF(3^3)
gap> w:= f^2;
( GF(3)^2 )
gap> map1:= LeftModuleGeneralMappingByImages( f, v, [ Z(3)^0 ], [ Z(27) ] );
[ Z(3)^0 ] -> [ Z(3^3) ]
gap> ImagesSource( map1 );
<vector space over GF(3), with 1 generator>
gap> PreImagesRange( map1 );
<vector space over GF(3), with 1 generator>
gap> CoKernelOfAdditiveGeneralMapping( map1 );
<vector space of dimension 0 over GF(3)>
gap> KernelOfAdditiveGeneralMapping( map1 );
<vector space of dimension 0 over GF(3)>
gap> IsSingleValued( map1 );
true
gap> IsInjective( map1 );
true
gap> ImagesRepresentative( map1, 0*Z(3)^0 );
0*Z(3)
gap> ImagesRepresentative( map1, Z(3)^0 );
Z(3^3)
gap> PreImagesRepresentative( map1, 0*Z(3) );
0*Z(3)
gap> PreImagesRepresentative( map1, Z(27) );
Z(3)^0
gap> 2 * map1 = - map1;
true
gap> Z(3) * map1 = - map1;
true
gap> map2:= LeftModuleGeneralMappingByImages( v, w,
> CanonicalBasis( v ),
> [ [ Z(3)^0, 0*Z(3)^0 ], [ Z(3)^0, Z(3)^0 ], [ 0*Z(3)^0, Z(3)^0 ] ] );
CanonicalBasis( GF(3^3) ) -> [ [ Z(3)^0, 0*Z(3) ], [ Z(3)^0, Z(3)^0 ],
[ 0*Z(3), Z(3)^0 ] ]
gap> ImagesSource( map2 );
<vector space over GF(3), with 3 generators>
gap> PreImagesRange( map2 );
GF(3^3)
gap> CoKernelOfAdditiveGeneralMapping( map2 );
<vector space of dimension 0 over GF(3)>
gap> KernelOfAdditiveGeneralMapping( map2 );
<vector space over GF(3), with 1 generator>
gap> IsSingleValued( map2 );
true
gap> IsInjective( map2 );
false
gap> ImagesRepresentative( map2, Z(27) );
[ Z(3)^0, Z(3)^0 ]
gap> ImagesRepresentative( map2, Z(9) );
fail
gap> PreImagesRepresentative( map2, [ Z(3)^0, Z(3)^0 ] );
Z(3^3)
gap> PreImagesRepresentative( map2, [ 0*Z(3)^0, 0*Z(3)^0 ] );
0*Z(3)
gap> 2 * map2 = - map2;
true
gap> Z(3) * map2 = - map2;
true
gap> map3:= LeftModuleGeneralMappingByImages( w, v,
> [ [ Z(3)^0, 0*Z(3)^0 ], [ Z(3)^0, Z(3)^0 ], [ 0*Z(3)^0, Z(3)^0 ] ],
> CanonicalBasis( v ) );
[ [ Z(3)^0, 0*Z(3) ], [ Z(3)^0, Z(3)^0 ], [ 0*Z(3), Z(3)^0 ]
] -> CanonicalBasis( GF(3^3) )
gap> ImagesSource( map3 );
GF(3^3)
gap> PreImagesRange( map3 );
<vector space over GF(3), with 3 generators>
gap> CoKernelOfAdditiveGeneralMapping( map3 );
<vector space over GF(3), with 1 generator>
gap> KernelOfAdditiveGeneralMapping( map3 );
<vector space of dimension 0 over GF(3)>
gap> IsSingleValued( map3 );
false
gap> IsInjective( map3 );
true
gap> ImagesRepresentative( map3, [ Z(3)^0, 0*Z(3)^0 ] );
Z(3)^0
gap> ImagesRepresentative( map3, [ 0*Z(3)^0, Z(3)^0 ] );
Z(3^3)^3
gap> PreImagesRepresentative( map3, Z(27) );
[ Z(3)^0, Z(3)^0 ]
gap> PreImagesRepresentative( map3, Z(3)^0 );
[ Z(3)^0, 0*Z(3) ]
gap> 2 * map3 = - map3;
true
gap> Z(3) * map3 = - map3;
true
gap> comp1:= CompositionMapping( map3, map2 );
CompositionMapping( [ [ Z(3)^0, 0*Z(3) ], [ Z(3)^0, Z(3)^0 ],
[ 0*Z(3), Z(3)^0 ] ] -> CanonicalBasis( GF(3^3) ),
CanonicalBasis( GF(3^3) ) -> [ [ Z(3)^0, 0*Z(3) ], [ Z(3)^0, Z(3)^0 ],
[ 0*Z(3), Z(3)^0 ] ] )
gap> IsInjective( comp1 );
false
gap> IsSingleValued( comp1 );
false
gap> IsSurjective( comp1 );
true
gap> comp2:= CompositionMapping( map2, map3 );
CompositionMapping( CanonicalBasis( GF(3^3) ) ->
[ [ Z(3)^0, 0*Z(3) ], [ Z(3)^0, Z(3)^0 ], [ 0*Z(3), Z(3)^0 ] ],
[ [ Z(3)^0, 0*Z(3) ], [ Z(3)^0, Z(3)^0 ], [ 0*Z(3), Z(3)^0 ]
] -> CanonicalBasis( GF(3^3) ) )
gap> IsInjective( comp2 );
true
gap> IsSingleValued( comp2 );
true
gap> IsSurjective( comp2 );
true
gap> comp3:= CompositionMapping( FrobeniusAutomorphism( v ), map1 );
[ Z(3)^0 ] -> [ Z(3^3)^3 ]
gap> IsInjective( comp3 );
true
gap> IsSingleValued( comp3 );
true
gap> IsSurjective( comp3 );
false
gap> 2 * comp3;
[ Z(3)^0 ] -> [ Z(3^3)^16 ]
gap> ImagesRepresentative( comp3, 0*Z(3)^0 );
0*Z(3)
gap> ImagesRepresentative( comp3, Z(3)^0 );
Z(3^3)^3
gap> sum:= map1 + map1;
[ Z(3)^0 ] -> [ Z(3^3)^14 ]
gap> sum + map1 = ZeroMapping( f, v );
true
gap> map4:= LeftModuleGeneralMappingByImages( v, v, CanonicalBasis( v ),
> [ Z(27)^8, Z(3), Z(27) ] );
CanonicalBasis( GF(3^3) ) -> [ Z(3^3)^8, Z(3), Z(3^3) ]
gap> map4 + IdentityMapping( v );
CanonicalBasis( GF(3^3) ) -> [ Z(3^3)^15, Z(3^3)^3, Z(3^3)^10 ]
gap> IdentityMapping( v ) + map4;
CanonicalBasis( GF(3^3) ) -> [ Z(3^3)^15, Z(3^3)^3, Z(3^3)^10 ]
# some tests involving zero mappings
gap> m:= GroupRing( GF(3), CyclicGroup( 2 ) );;
gap> t:= TrivialSubspace( m );;
gap> bm:= BasisVectors( Basis( m ) );;
gap> bt:= BasisVectors( Basis( t ) );;
gap> funs:= [ IsInjective, IsSurjective, IsTotal, IsSingleValued ];;
gap> map:= LeftModuleGeneralMappingByImages( m, m, bm, 0 * bm );;
gap> List( funs, f -> f( map ) );
[ false, false, true, true ]
gap> map:= LeftModuleGeneralMappingByImages( m, t, bm, 0 * bm );;
gap> List( funs, f -> f( map ) );
[ false, true, true, true ]
gap> map:= LeftModuleGeneralMappingByImages( t, t, bt, 0 * bt );;
gap> List( funs, f -> f( map ) );
[ true, true, true, true ]
gap> map:= LeftModuleGeneralMappingByImages( t, m, bt, 0 * bt );;
gap> List( funs, f -> f( map ) );
[ true, false, true, true ]
gap> map:= LeftModuleGeneralMappingByImages( m, m, 0 * bm, bm );;
gap> List( funs, f -> f( map ) );
[ true, true, false, false ]
gap> map:= LeftModuleGeneralMappingByImages( m, t, bt, 0 * bt );;
gap> List( funs, f -> f( map ) );
[ true, true, false, true ]
gap> map:= LeftModuleGeneralMappingByImages( t, m, 0 * bm, bm );;
gap> List( funs, f -> f( map ) );
[ true, true, true, false ]
#############################################################################
##
## tests for linear mappings given by matrices
## (same tests as above,
## except those that would involve matrices with zero rows or columns)
##
gap> bf:= CanonicalBasis( GF(3) );
CanonicalBasis( GF(3) )
gap> bv:= CanonicalBasis( GF(27) );
CanonicalBasis( GF(3^3) )
gap> bw:= CanonicalBasis( f^2 );
CanonicalBasis( ( GF(3)^2 ) )
gap> map5:= LeftModuleHomomorphismByMatrix( bf,
> [ [ Z(3), Z(3), Z(3) ] ], bv );
<linear mapping by matrix, GF(3) -> GF(3^3)>
gap> ImagesSource( map5 );
<vector space over GF(3), with 1 generator>
gap> PreImagesRange( map5 );
GF(3)
gap> CoKernelOfAdditiveGeneralMapping( map5 );
<algebra of dimension 0 over GF(3)>
gap> KernelOfAdditiveGeneralMapping( map5 );
<vector space of dimension 0 over GF(3)>
gap> IsSingleValued( map5 );
true
gap> IsInjective( map5 );
true
gap> ImagesRepresentative( map5, 0*Z(3)^0 );
0*Z(3)
gap> ImagesRepresentative( map5, Z(3)^0 );
Z(3^3)^19
gap> PreImagesRepresentative( map5, 0*Z(3) );
0*Z(3)
gap> PreImagesRepresentative( map5, Z(27) );
fail
gap> 2 * map5 = - map5;
true
gap> Z(3) * map5 = - map5;
true
gap> map6:= LeftModuleHomomorphismByMatrix( bv,
> [ [ Z(3), Z(3) ], [ 0*Z(3), 0*Z(3) ], [ Z(3), 0*Z(3) ] ],
> bw );
<linear mapping by matrix, GF(3^3) -> ( GF(3)^2 )>
gap> ImagesSource( map6 );
<vector space over GF(3), with 3 generators>
gap> PreImagesRange( map6 );
GF(3^3)
gap> CoKernelOfAdditiveGeneralMapping( map6 );
<vector space of dimension 0 over GF(3)>
gap> KernelOfAdditiveGeneralMapping( map6 );
<vector space over GF(3), with 1 generator>
gap> IsSingleValued( map6 );
true
gap> IsInjective( map6 );
false
gap> ImagesRepresentative( map6, Z(27) );
[ 0*Z(3), 0*Z(3) ]
gap> ImagesRepresentative( map6, Z(9) );
fail
gap> PreImagesRepresentative( map6, [ Z(3)^0, Z(3)^0 ] );
Z(3)
gap> PreImagesRepresentative( map6, [ 0*Z(3)^0, 0*Z(3)^0 ] );
0*Z(3)
gap> 2 * map6 = - map6;
true
gap> Z(3) * map6 = - map6;
true
gap> map7:= LeftModuleHomomorphismByMatrix( bw,
> [ [ Z(3)^0, 0*Z(3)^0, Z(3)^0 ], [ Z(3)^0, 0*Z(3)^0, Z(3)^0 ] ],
> bv );
<linear mapping by matrix, ( GF(3)^2 ) -> GF(3^3)>
gap> ImagesSource( map7 );
<vector space over GF(3), with 2 generators>
gap> PreImagesRange( map7 );
( GF(3)^2 )
gap> CoKernelOfAdditiveGeneralMapping( map7 );
<algebra of dimension 0 over GF(3)>
gap> KernelOfAdditiveGeneralMapping( map7 );
<vector space over GF(3), with 1 generator>
gap> IsSingleValued( map7 );
true
gap> IsInjective( map7 );
false
gap> ImagesRepresentative( map7, [ Z(3)^0, 0*Z(3)^0 ] );
Z(3^3)^21
gap> ImagesRepresentative( map7, [ 0*Z(3)^0, Z(3)^0 ] );
Z(3^3)^21
gap> PreImagesRepresentative( map7, Z(27) );
fail
gap> PreImagesRepresentative( map7, Z(3)^0 );
fail
gap> 2 * map7 = - map7;
true
gap> Z(3) * map7 = - map7;
true
gap> comp1:= CompositionMapping( map7, map6 );
<linear mapping by matrix, GF(3^3) -> GF(3^3)>
gap> IsInjective( comp1 );
false
gap> IsSingleValued( comp1 );
true
gap> IsSurjective( comp1 );
false
gap> comp2:= CompositionMapping( map6, map7 );
<linear mapping by matrix, ( GF(3)^2 ) -> ( GF(3)^2 )>
gap> IsInjective( comp2 );
false
gap> IsSingleValued( comp2 );
true
gap> IsSurjective( comp2 );
false
gap> comp3:= CompositionMapping( FrobeniusAutomorphism( v ), map5 );
<linear mapping by matrix, GF(3) -> GF(3^3)>
gap> IsInjective( comp3 );
true
gap> IsSingleValued( comp3 );
true
gap> IsSurjective( comp3 );
false
gap> 2 * comp3;
<linear mapping by matrix, GF(3) -> GF(3^3)>
gap> ImagesRepresentative( comp3, 0*Z(3)^0 );
0*Z(3)
gap> ImagesRepresentative( comp3, Z(3)^0 );
Z(3^3)^5
gap> sum:= map1 + map1;
[ Z(3)^0 ] -> [ Z(3^3)^14 ]
gap> sum + map1 = ZeroMapping( f, v );
true
gap> map8:= LeftModuleHomomorphismByMatrix( bv,
> [ [ Z(3), Z(3), Z(3) ],
> [ 0*Z(3), 0*Z(3), Z(3)^0 ],
> [ Z(3), 0*Z(3), 0*Z(3) ] ],
> bv );
<linear mapping by matrix, GF(3^3) -> GF(3^3)>
gap> map8 + IdentityMapping( v );
<linear mapping by matrix, GF(3^3) -> GF(3^3)>
gap> IdentityMapping( v ) + map8;
<linear mapping by matrix, GF(3^3) -> GF(3^3)>
#############################################################################
##
## tests for mixed cases
##
gap> map2 + map6;
CanonicalBasis( GF(3^3) ) -> [ [ 0*Z(3), Z(3) ], [ Z(3)^0, Z(3)^0 ],
[ Z(3), Z(3)^0 ] ]
gap> map6 + map2;
CanonicalBasis( GF(3^3) ) -> [ [ 0*Z(3), Z(3) ], [ Z(3)^0, Z(3)^0 ],
[ Z(3), Z(3)^0 ] ]
gap> # id:= IdentityMapping( v );
gap> # 2 * id;
gap> # id + id;
gap> # - id;
gap> # zero:= ZeroMapping( v, v );
gap> # 2 * zero;
gap> # zero + zero;
gap> # id + zero;
gap> # - zero;
#############################################################################
##
## tests for natural homomorphisms
##
gap> NaturalHomomorphismBySubspace( v, TrivialSubspace( v ) )
> = IdentityMapping( v );
true
gap> IsZero( NaturalHomomorphismBySubspace( v, v ) );
true
gap> nathom:= NaturalHomomorphismBySubspace( w,
> Subspace( w, [ [ Z(3), Z(3) ] ] ) );
<linear mapping by matrix, ( GF(3)^2 ) -> ( GF(3)^1 )>
gap> ImagesSource( nathom );
( GF(3)^1 )
gap> PreImagesRange( nathom );
( GF(3)^2 )
gap> CoKernelOfAdditiveGeneralMapping( nathom );
<vector space of dimension 0 over GF(3)>
gap> KernelOfAdditiveGeneralMapping( nathom );
<vector space of dimension 1 over GF(3)>
gap> IsInjective( nathom );
false
gap> IsSingleValued( nathom );
true
gap> IsSurjective( nathom );
true
gap> IsTotal( nathom );
true
gap> ImagesRepresentative( nathom, [ Z(3), Z(3)^0 ] );
[ Z(3)^0 ]
gap> ImagesRepresentative( nathom, [ Z(3)^0, Z(3)^0 ] );
[ 0*Z(3) ]
gap> PreImagesRepresentative( nathom, [ Z(3)^0 ] );
[ Z(3)^0, 0*Z(3) ]
gap> PreImagesRepresentative( nathom, [ 0*Z(3) ] );
[ 0*Z(3), 0*Z(3) ]
gap> 2 * nathom = - nathom;
true
gap> Z(3) * nathom = - nathom;
true
#############################################################################
##
## tests for spaces of linear mappings
##
gap> hom:= Hom( f, v, w );
Hom( GF(3), GF(3^3), ( GF(3)^2 ) )
gap> IsFullHomModule( hom );
true
gap> Dimension( hom );
6
gap> Random( hom ) in hom;
true
gap> map6 in hom;
true
gap> sub:= Subspace( hom, [ map6 ] );
<vector space over GF(3), with 1 generator>
gap> BasisVectors( Basis( sub ) );
[ <linear mapping by matrix, GF(3^3) -> ( GF(3)^2 )> ]
gap> sub:= LeftModuleByGenerators( f, [ map6 ] );
<vector space over GF(3), with 1 generator>
gap> Dimension( sub );
1
gap> zero:= ZeroMapping( v, w );
ZeroMapping( GF(3^3), ( GF(3)^2 ) )
gap> triv:= LeftModuleByGenerators( f, [], zero );
<vector space of dimension 0 over GF(3)>
gap> IsSubset( hom, triv );
true
gap> mb:= MutableBasis( f, [], zero );
<mutable basis over GF(3), 0 vectors>
gap> CloseMutableBasis( mb, zero );
false
gap> CloseMutableBasis( mb, map6 );
true
gap> ImmutableBasis( mb );
Basis( <vector space of dimension 1 over GF(3)>, ... )
#############################################################################
##
## tests for algebras of linear mappings
##
gap> endo:= End( f, v );
End( GF(3), GF(3^3) )
gap> id:= IdentityMapping( v );
IdentityMapping( GF(3^3) )
gap> zero:= ZeroMapping( v, v );
ZeroMapping( GF(3^3), GF(3^3) )
gap> id in endo;
true
gap> zero in endo;
true
gap> RingByGenerators( [ id ] );
<algebra over GF(3), with 1 generator>
gap> DefaultRingByGenerators( [ id ] );
<algebra over GF(3), with 1 generator>
gap> RingWithOneByGenerators( [ id ] );
<algebra-with-one over GF(3), with 1 generator>
gap> a:= AlgebraByGenerators( f, [], zero );
<algebra of dimension 0 over GF(3)>
gap> Dimension( a );
0
gap> a:= AlgebraByGenerators( f, [ id ] );
<algebra over GF(3), with 1 generator>
gap> Dimension( a );
1
gap> a:= AlgebraWithOneByGenerators( f, [], zero );
<algebra-with-one over GF(3), with 0 generators>
gap> Dimension( a );
1
gap> a:= AlgebraWithOneByGenerators( f, [ id ] );
<algebra-with-one over GF(3), with 1 generator>
gap> Dimension( a );
1
gap> IsFullHomModule( a );
false
gap> IsFullHomModule( endo );
true
gap> Dimension( endo );
9
gap> Random( endo ) in endo;
true
gap> GeneratorsOfLeftModule( endo );
[ <linear mapping by matrix, GF(3^3) -> GF(3^3)>,
<linear mapping by matrix, GF(3^3) -> GF(3^3)>,
<linear mapping by matrix, GF(3^3) -> GF(3^3)>,
<linear mapping by matrix, GF(3^3) -> GF(3^3)>,
<linear mapping by matrix, GF(3^3) -> GF(3^3)>,
<linear mapping by matrix, GF(3^3) -> GF(3^3)>,
<linear mapping by matrix, GF(3^3) -> GF(3^3)>,
<linear mapping by matrix, GF(3^3) -> GF(3^3)>,
<linear mapping by matrix, GF(3^3) -> GF(3^3)> ]
gap> b:= Basis( endo );
Basis( End( GF(3), GF(3^3) ),
[ <linear mapping by matrix, GF(3^3) -> GF(3^3)>,
<linear mapping by matrix, GF(3^3) -> GF(3^3)>,
<linear mapping by matrix, GF(3^3) -> GF(3^3)>,
<linear mapping by matrix, GF(3^3) -> GF(3^3)>,
<linear mapping by matrix, GF(3^3) -> GF(3^3)>,
<linear mapping by matrix, GF(3^3) -> GF(3^3)>,
<linear mapping by matrix, GF(3^3) -> GF(3^3)>,
<linear mapping by matrix, GF(3^3) -> GF(3^3)>,
<linear mapping by matrix, GF(3^3) -> GF(3^3)> ] )
gap> BasisVectors( b );
[ <linear mapping by matrix, GF(3^3) -> GF(3^3)>,
<linear mapping by matrix, GF(3^3) -> GF(3^3)>,
<linear mapping by matrix, GF(3^3) -> GF(3^3)>,
<linear mapping by matrix, GF(3^3) -> GF(3^3)>,
<linear mapping by matrix, GF(3^3) -> GF(3^3)>,
<linear mapping by matrix, GF(3^3) -> GF(3^3)>,
<linear mapping by matrix, GF(3^3) -> GF(3^3)>,
<linear mapping by matrix, GF(3^3) -> GF(3^3)>,
<linear mapping by matrix, GF(3^3) -> GF(3^3)> ]
gap> Coefficients( b, id );
[ Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0 ]
gap> map:= LeftModuleHomomorphismByMatrix( bv, 2 * IdentityMat( 3, f ), bv );
<linear mapping by matrix, GF(3^3) -> GF(3^3)>
gap> Coefficients( b, map );
[ Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3) ]
gap> endoendo:= End( f, endo );
End( GF(3), End( GF(3), GF(3^3) ) )
gap> Dimension( endoendo );
81
gap> STOP_TEST("vspchom.tst");
[ Dauer der Verarbeitung: 0.24 Sekunden
(vorverarbeitet)
]
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