File: vspchom.gd

package info (click to toggle)
gap 4r4p12-2
  • links: PTS
  • area: main
  • in suites: squeeze, wheezy
  • size: 29,584 kB
  • ctags: 7,113
  • sloc: ansic: 98,786; sh: 3,299; perl: 2,263; makefile: 498; asm: 63; awk: 6
file content (213 lines) | stat: -rw-r--r-- 8,149 bytes parent folder | download | duplicates (4)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
#############################################################################
##
#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 $";


#############################################################################
#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 );


#############################################################################
##
#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 ] );


#############################################################################
##
#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