This file is indexed.

/usr/share/gap/doc/ref/chap57.html is in gap-doc 4r8p6-2.

This file is owned by root:root, with mode 0o644.

The actual contents of the file can be viewed below.

  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
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
<?xml version="1.0" encoding="UTF-8"?>

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
         "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">

<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head>
<title>GAP (ref) - Chapter 57: Modules</title>
<meta http-equiv="content-type" content="text/html; charset=UTF-8" />
<meta name="generator" content="GAPDoc2HTML" />
<link rel="stylesheet" type="text/css" href="manual.css" />
<script src="manual.js" type="text/javascript"></script>
<script type="text/javascript">overwriteStyle();</script>
</head>
<body class="chap57"  onload="jscontent()">


<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a>  <a href="chap1.html">1</a>  <a href="chap2.html">2</a>  <a href="chap3.html">3</a>  <a href="chap4.html">4</a>  <a href="chap5.html">5</a>  <a href="chap6.html">6</a>  <a href="chap7.html">7</a>  <a href="chap8.html">8</a>  <a href="chap9.html">9</a>  <a href="chap10.html">10</a>  <a href="chap11.html">11</a>  <a href="chap12.html">12</a>  <a href="chap13.html">13</a>  <a href="chap14.html">14</a>  <a href="chap15.html">15</a>  <a href="chap16.html">16</a>  <a href="chap17.html">17</a>  <a href="chap18.html">18</a>  <a href="chap19.html">19</a>  <a href="chap20.html">20</a>  <a href="chap21.html">21</a>  <a href="chap22.html">22</a>  <a href="chap23.html">23</a>  <a href="chap24.html">24</a>  <a href="chap25.html">25</a>  <a href="chap26.html">26</a>  <a href="chap27.html">27</a>  <a href="chap28.html">28</a>  <a href="chap29.html">29</a>  <a href="chap30.html">30</a>  <a href="chap31.html">31</a>  <a href="chap32.html">32</a>  <a href="chap33.html">33</a>  <a href="chap34.html">34</a>  <a href="chap35.html">35</a>  <a href="chap36.html">36</a>  <a href="chap37.html">37</a>  <a href="chap38.html">38</a>  <a href="chap39.html">39</a>  <a href="chap40.html">40</a>  <a href="chap41.html">41</a>  <a href="chap42.html">42</a>  <a href="chap43.html">43</a>  <a href="chap44.html">44</a>  <a href="chap45.html">45</a>  <a href="chap46.html">46</a>  <a href="chap47.html">47</a>  <a href="chap48.html">48</a>  <a href="chap49.html">49</a>  <a href="chap50.html">50</a>  <a href="chap51.html">51</a>  <a href="chap52.html">52</a>  <a href="chap53.html">53</a>  <a href="chap54.html">54</a>  <a href="chap55.html">55</a>  <a href="chap56.html">56</a>  <a href="chap57.html">57</a>  <a href="chap58.html">58</a>  <a href="chap59.html">59</a>  <a href="chap60.html">60</a>  <a href="chap61.html">61</a>  <a href="chap62.html">62</a>  <a href="chap63.html">63</a>  <a href="chap64.html">64</a>  <a href="chap65.html">65</a>  <a href="chap66.html">66</a>  <a href="chap67.html">67</a>  <a href="chap68.html">68</a>  <a href="chap69.html">69</a>  <a href="chap70.html">70</a>  <a href="chap71.html">71</a>  <a href="chap72.html">72</a>  <a href="chap73.html">73</a>  <a href="chap74.html">74</a>  <a href="chap75.html">75</a>  <a href="chap76.html">76</a>  <a href="chap77.html">77</a>  <a href="chap78.html">78</a>  <a href="chap79.html">79</a>  <a href="chap80.html">80</a>  <a href="chap81.html">81</a>  <a href="chap82.html">82</a>  <a href="chap83.html">83</a>  <a href="chap84.html">84</a>  <a href="chap85.html">85</a>  <a href="chap86.html">86</a>  <a href="chap87.html">87</a>  <a href="chapBib.html">Bib</a>  <a href="chapInd.html">Ind</a>  </div>

<div class="chlinkprevnexttop">&nbsp;<a href="chap0.html">[Top of Book]</a>&nbsp;  <a href="chap0.html#contents">[Contents]</a>&nbsp;  &nbsp;<a href="chap56.html">[Previous Chapter]</a>&nbsp;  &nbsp;<a href="chap58.html">[Next Chapter]</a>&nbsp;  </div>

<p id="mathjaxlink" class="pcenter"><a href="chap57_mj.html">[MathJax on]</a></p>
<p><a id="X8183A6857B0C3633" name="X8183A6857B0C3633"></a></p>
<div class="ChapSects"><a href="chap57.html#X8183A6857B0C3633">57 <span class="Heading">Modules</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap57.html#X87A33EFD7CC179C1">57.1 <span class="Heading">Generating modules</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap57.html#X7C62FE5282E9C505">57.1-1 IsLeftOperatorAdditiveGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap57.html#X7ED323027B291BDF">57.1-2 IsLeftModule</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap57.html#X7F76B1FD84775025">57.1-3 GeneratorsOfLeftOperatorAdditiveGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap57.html#X7C7684EF867323C2">57.1-4 GeneratorsOfLeftModule</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap57.html#X7EB3E46D7BC4A35C">57.1-5 AsLeftModule</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap57.html#X7F19AD3D799D0469">57.1-6 IsRightOperatorAdditiveGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap57.html#X8479A5AA7DF25F50">57.1-7 IsRightModule</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap57.html#X7DBC4BCB876EEE1C">57.1-8 GeneratorsOfRightOperatorAdditiveGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap57.html#X8586A83B85F176F6">57.1-9 GeneratorsOfRightModule</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap57.html#X79ED1D7D7F0AE59A">57.1-10 LeftModuleByGenerators</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap57.html#X86F070E0807DC34E">57.1-11 LeftActingDomain</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap57.html#X7934FAE97B6D2AD8">57.2 <span class="Heading">Submodules</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap57.html#X8465103F874BC07B">57.2-1 Submodule</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap57.html#X83CF3AD18050C982">57.2-2 SubmoduleNC</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap57.html#X7C68C4E287481EC0">57.2-3 ClosureLeftModule</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap57.html#X7980BC20856B2B7D">57.2-4 TrivialSubmodule</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap57.html#X85BD57F27F513D3E">57.3 <span class="Heading">Free Modules</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap57.html#X7C4832187F3D9228">57.3-1 IsFreeLeftModule</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap57.html#X7C043E307E344AEE">57.3-2 FreeLeftModule</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap57.html#X7E6926C6850E7C4E">57.3-3 Dimension</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap57.html#X802DB9FB824B0167">57.3-4 IsFiniteDimensional</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap57.html#X7909E8E785420F0E">57.3-5 UseBasis</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap57.html#X7C8F844783F4FA09">57.3-6 IsRowModule</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap57.html#X81FCC1D780435CF1">57.3-7 IsMatrixModule</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap57.html#X853E085C868196EF">57.3-8 IsFullRowModule</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap57.html#X848041A47BC4B038">57.3-9 FullRowModule</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap57.html#X814CEA62842CF5BB">57.3-10 IsFullMatrixModule</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap57.html#X7A0C871B7C446F1F">57.3-11 FullMatrixModule</a></span>
</div></div>
</div>

<h3>57 <span class="Heading">Modules</span></h3>

<p><a id="X87A33EFD7CC179C1" name="X87A33EFD7CC179C1"></a></p>

<h4>57.1 <span class="Heading">Generating modules</span></h4>

<p><a id="X7C62FE5282E9C505" name="X7C62FE5282E9C505"></a></p>

<h5>57.1-1 IsLeftOperatorAdditiveGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsLeftOperatorAdditiveGroup</code>( <var class="Arg">D</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>A domain <var class="Arg">D</var> lies in <code class="code">IsLeftOperatorAdditiveGroup</code> if it is an additive group that is closed under scalar multiplication from the left, and such that <span class="SimpleMath">λ * ( x + y ) = λ * x + λ * y</span> for all scalars <span class="SimpleMath">λ</span> and elements <span class="SimpleMath">x, y ∈ D</span> (here and below by scalars we mean elements of a domain acting on <var class="Arg">D</var> from left or right as appropriate).</p>

<p><a id="X7ED323027B291BDF" name="X7ED323027B291BDF"></a></p>

<h5>57.1-2 IsLeftModule</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsLeftModule</code>( <var class="Arg">M</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>A domain <var class="Arg">M</var> lies in <code class="code">IsLeftModule</code> if it lies in <code class="code">IsLeftOperatorAdditiveGroup</code>, <em>and</em> the set of scalars forms a ring, <em>and</em> <span class="SimpleMath">(λ + μ) * x = λ * x + μ * x</span> for scalars <span class="SimpleMath">λ, μ</span> and <span class="SimpleMath">x ∈ M</span>, <em>and</em> scalar multiplication satisfies <span class="SimpleMath">λ * (μ * x) = (λ * μ) * x</span> for scalars <span class="SimpleMath">λ, μ</span> and <span class="SimpleMath">x ∈ M</span>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">V:= FullRowSpace( Rationals, 3 );</span>
( Rationals^3 )
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">IsLeftModule( V );</span>
true
</pre></div>

<p><a id="X7F76B1FD84775025" name="X7F76B1FD84775025"></a></p>

<h5>57.1-3 GeneratorsOfLeftOperatorAdditiveGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; GeneratorsOfLeftOperatorAdditiveGroup</code>( <var class="Arg">D</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a list of elements of <var class="Arg">D</var> that generates <var class="Arg">D</var> as a left operator additive group.</p>

<p><a id="X7C7684EF867323C2" name="X7C7684EF867323C2"></a></p>

<h5>57.1-4 GeneratorsOfLeftModule</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; GeneratorsOfLeftModule</code>( <var class="Arg">M</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a list of elements of <var class="Arg">M</var> that generate <var class="Arg">M</var> as a left module.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">V:= FullRowSpace( Rationals, 3 );;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">GeneratorsOfLeftModule( V );</span>
[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ]
</pre></div>

<p><a id="X7EB3E46D7BC4A35C" name="X7EB3E46D7BC4A35C"></a></p>

<h5>57.1-5 AsLeftModule</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; AsLeftModule</code>( <var class="Arg">R</var>, <var class="Arg">D</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>if the domain <var class="Arg">D</var> forms an additive group and is closed under left multiplication by the elements of <var class="Arg">R</var>, then <code class="code">AsLeftModule( <var class="Arg">R</var>, <var class="Arg">D</var> )</code> returns the domain <var class="Arg">D</var> viewed as a left module.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">coll:= [[0*Z(2),0*Z(2)], [Z(2),0*Z(2)], [0*Z(2),Z(2)], [Z(2),Z(2)]];</span>
[ [ 0*Z(2), 0*Z(2) ], [ Z(2)^0, 0*Z(2) ], [ 0*Z(2), Z(2)^0 ], 
  [ Z(2)^0, Z(2)^0 ] ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">AsLeftModule( GF(2), coll );</span>
&lt;vector space of dimension 2 over GF(2)&gt;
</pre></div>

<p><a id="X7F19AD3D799D0469" name="X7F19AD3D799D0469"></a></p>

<h5>57.1-6 IsRightOperatorAdditiveGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsRightOperatorAdditiveGroup</code>( <var class="Arg">D</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>A domain <var class="Arg">D</var> lies in <code class="code">IsRightOperatorAdditiveGroup</code> if it is an additive group that is closed under scalar multiplication from the right, and such that <span class="SimpleMath">( x + y ) * λ = x * λ + y * λ</span> for all scalars <span class="SimpleMath">λ</span> and elements <span class="SimpleMath">x, y ∈ D</span>.</p>

<p><a id="X8479A5AA7DF25F50" name="X8479A5AA7DF25F50"></a></p>

<h5>57.1-7 IsRightModule</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsRightModule</code>( <var class="Arg">M</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>A domain <var class="Arg">M</var> lies in <code class="code">IsRightModule</code> if it lies in <code class="code">IsRightOperatorAdditiveGroup</code>, <em>and</em> the set of scalars forms a ring, <em>and</em> <span class="SimpleMath">x * (λ + μ) = x * λ + x * μ</span> for scalars <span class="SimpleMath">λ, μ</span> and <span class="SimpleMath">x ∈ M</span>, <em>and</em> scalar multiplication satisfies <span class="SimpleMath">(x * μ) * λ = x * (μ * λ)</span> for scalars <span class="SimpleMath">λ, μ</span> and <span class="SimpleMath">x ∈ M</span>.</p>

<p><a id="X7DBC4BCB876EEE1C" name="X7DBC4BCB876EEE1C"></a></p>

<h5>57.1-8 GeneratorsOfRightOperatorAdditiveGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; GeneratorsOfRightOperatorAdditiveGroup</code>( <var class="Arg">D</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a list of elements of <var class="Arg">D</var> that generates <var class="Arg">D</var> as a right operator additive group.</p>

<p><a id="X8586A83B85F176F6" name="X8586A83B85F176F6"></a></p>

<h5>57.1-9 GeneratorsOfRightModule</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; GeneratorsOfRightModule</code>( <var class="Arg">M</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a list of elements of <var class="Arg">M</var> that generate <var class="Arg">M</var> as a left module.</p>

<p><a id="X79ED1D7D7F0AE59A" name="X79ED1D7D7F0AE59A"></a></p>

<h5>57.1-10 LeftModuleByGenerators</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; LeftModuleByGenerators</code>( <var class="Arg">R</var>, <var class="Arg">gens</var>[, <var class="Arg">zero</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns the left module over <var class="Arg">R</var> generated by <var class="Arg">gens</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">coll:= [ [Z(2),0*Z(2)], [0*Z(2),Z(2)], [Z(2),Z(2)] ];;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">V:= LeftModuleByGenerators( GF(16), coll );</span>
&lt;vector space over GF(2^4), with 3 generators&gt;
</pre></div>

<p><a id="X86F070E0807DC34E" name="X86F070E0807DC34E"></a></p>

<h5>57.1-11 LeftActingDomain</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; LeftActingDomain</code>( <var class="Arg">D</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Let <var class="Arg">D</var> be an external left set, that is, <var class="Arg">D</var> is closed under the action of a domain <span class="SimpleMath">L</span> by multiplication from the left. Then <span class="SimpleMath">L</span> can be accessed as value of <code class="code">LeftActingDomain</code> for <var class="Arg">D</var>.</p>

<p><a id="X7934FAE97B6D2AD8" name="X7934FAE97B6D2AD8"></a></p>

<h4>57.2 <span class="Heading">Submodules</span></h4>

<p><a id="X8465103F874BC07B" name="X8465103F874BC07B"></a></p>

<h5>57.2-1 Submodule</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Submodule</code>( <var class="Arg">M</var>, <var class="Arg">gens</var>[, <var class="Arg">"basis"</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>is the left module generated by the collection <var class="Arg">gens</var>, with parent module <var class="Arg">M</var>. If the string <code class="code">"basis"</code> is entered as the third argument then the submodule of <var class="Arg">M</var> is created for which the list <var class="Arg">gens</var> is known to be a list of basis vectors; in this case, it is <em>not</em> checked whether <var class="Arg">gens</var> really is linearly independent and whether all in <var class="Arg">gens</var> lie in <var class="Arg">M</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">coll:= [ [Z(2),0*Z(2)], [0*Z(2),Z(2)], [Z(2),Z(2)] ];;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">V:= LeftModuleByGenerators( GF(16), coll );;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">W:= Submodule( V, [ coll[1], coll[2] ] );</span>
&lt;vector space over GF(2^4), with 2 generators&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Parent( W ) = V;</span>
true
</pre></div>

<p><a id="X83CF3AD18050C982" name="X83CF3AD18050C982"></a></p>

<h5>57.2-2 SubmoduleNC</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; SubmoduleNC</code>( <var class="Arg">M</var>, <var class="Arg">gens</var>[, <var class="Arg">"basis"</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">SubmoduleNC</code> does the same as <code class="func">Submodule</code> (<a href="chap57.html#X8465103F874BC07B"><span class="RefLink">57.2-1</span></a>), except that it does not check whether all in <var class="Arg">gens</var> lie in <var class="Arg">M</var>.</p>

<p><a id="X7C68C4E287481EC0" name="X7C68C4E287481EC0"></a></p>

<h5>57.2-3 ClosureLeftModule</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ClosureLeftModule</code>( <var class="Arg">M</var>, <var class="Arg">m</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>is the left module generated by the left module generators of <var class="Arg">M</var> and the element <var class="Arg">m</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">V:= LeftModuleByGenerators(Rationals, [ [ 1, 0, 0 ], [ 0, 1, 0 ] ]);</span>
&lt;vector space over Rationals, with 2 generators&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">ClosureLeftModule( V, [ 1, 1, 1 ] );</span>
&lt;vector space over Rationals, with 3 generators&gt;
</pre></div>

<p><a id="X7980BC20856B2B7D" name="X7980BC20856B2B7D"></a></p>

<h5>57.2-4 TrivialSubmodule</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; TrivialSubmodule</code>( <var class="Arg">M</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the zero submodule of <var class="Arg">M</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">V:= LeftModuleByGenerators(Rationals, [[ 1, 0, 0 ], [ 0, 1, 0 ]]);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">TrivialSubmodule( V );</span>
&lt;vector space over Rationals, with 0 generators&gt;
</pre></div>

<p><a id="X85BD57F27F513D3E" name="X85BD57F27F513D3E"></a></p>

<h4>57.3 <span class="Heading">Free Modules</span></h4>

<p><a id="X7C4832187F3D9228" name="X7C4832187F3D9228"></a></p>

<h5>57.3-1 IsFreeLeftModule</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsFreeLeftModule</code>( <var class="Arg">M</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>A left module is free as module if it is isomorphic to a direct sum of copies of its left acting domain.</p>

<p>Free left modules can have bases.</p>

<p>The characteristic (see <code class="func">Characteristic</code> (<a href="chap31.html#X81278E53800BF64D"><span class="RefLink">31.10-1</span></a>)) of a free left module is defined as the characteristic of its left acting domain (see <code class="func">LeftActingDomain</code> (<a href="chap57.html#X86F070E0807DC34E"><span class="RefLink">57.1-11</span></a>)).</p>

<p><a id="X7C043E307E344AEE" name="X7C043E307E344AEE"></a></p>

<h5>57.3-2 FreeLeftModule</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; FreeLeftModule</code>( <var class="Arg">R</var>, <var class="Arg">gens</var>[, <var class="Arg">zero</var>][, <var class="Arg">"basis"</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="code">FreeLeftModule( <var class="Arg">R</var>, <var class="Arg">gens</var> )</code> is the free left module over the ring <var class="Arg">R</var>, generated by the vectors in the collection <var class="Arg">gens</var>.</p>

<p>If there are three arguments, a ring <var class="Arg">R</var> and a collection <var class="Arg">gens</var> and an element <var class="Arg">zero</var>, then <code class="code">FreeLeftModule( <var class="Arg">R</var>, <var class="Arg">gens</var>, <var class="Arg">zero</var> )</code> is the <var class="Arg">R</var>-free left module generated by <var class="Arg">gens</var>, with zero element <var class="Arg">zero</var>.</p>

<p>If the last argument is the string <code class="code">"basis"</code> then the vectors in <var class="Arg">gens</var> are known to form a basis of the free module.</p>

<p>It should be noted that the generators <var class="Arg">gens</var> must be vectors, that is, they must support an addition and a scalar action of <var class="Arg">R</var> via left multiplication. (See also Section <a href="chap31.html#X82039A218274826F"><span class="RefLink">31.3</span></a> for the general meaning of "generators" in <strong class="pkg">GAP</strong>.) In particular, <code class="func">FreeLeftModule</code> is <em>not</em> an equivalent of commands such as <code class="func">FreeGroup</code> (<a href="chap37.html#X8215999E835290F0"><span class="RefLink">37.2-1</span></a>) in the sense of a constructor of a free group on abstract generators. Such a construction seems to be unnecessary for vector spaces, for that one can use for example row spaces (see <code class="func">FullRowSpace</code> (<a href="chap61.html#X80209A8785126AAB"><span class="RefLink">61.9-4</span></a>)) in the finite dimensional case and polynomial rings (see <code class="func">PolynomialRing</code> (<a href="chap66.html#X7D2F16E480060330"><span class="RefLink">66.15-1</span></a>)) in the infinite dimensional case. Moreover, the definition of a "natural" addition for elements of a given magma (for example a permutation group) is possible via the construction of magma rings (see Chapter <a href="chap65.html#X825897DC7A16E07D"><span class="RefLink">65</span></a>).</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">V:= FreeLeftModule(Rationals, [[ 1, 0, 0 ], [ 0, 1, 0 ]], "basis");</span>
&lt;vector space of dimension 2 over Rationals&gt;
</pre></div>

<p><a id="X7E6926C6850E7C4E" name="X7E6926C6850E7C4E"></a></p>

<h5>57.3-3 Dimension</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Dimension</code>( <var class="Arg">M</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>A free left module has dimension <span class="SimpleMath">n</span> if it is isomorphic to a direct sum of <span class="SimpleMath">n</span> copies of its left acting domain.</p>

<p>(We do <em>not</em> mark <code class="func">Dimension</code> as invariant under isomorphisms since we want to call <code class="func">UseIsomorphismRelation</code> (<a href="chap31.html#X839BE6467E8474D9"><span class="RefLink">31.13-3</span></a>) also for free left modules over different left acting domains.)</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">V:= FreeLeftModule( Rationals, [ [ 1, 0 ], [ 0, 1 ], [ 1, 1 ] ] );;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Dimension( V );</span>
2
</pre></div>

<p><a id="X802DB9FB824B0167" name="X802DB9FB824B0167"></a></p>

<h5>57.3-4 IsFiniteDimensional</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsFiniteDimensional</code>( <var class="Arg">M</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>is <code class="keyw">true</code> if <var class="Arg">M</var> is a free left module that is finite dimensional over its left acting domain, and <code class="keyw">false</code> otherwise.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">V:= FreeLeftModule( Rationals, [ [ 1, 0 ], [ 0, 1 ], [ 1, 1 ] ] );;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">IsFiniteDimensional( V );</span>
true
</pre></div>

<p><a id="X7909E8E785420F0E" name="X7909E8E785420F0E"></a></p>

<h5>57.3-5 UseBasis</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; UseBasis</code>( <var class="Arg">V</var>, <var class="Arg">gens</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>The vectors in the list <var class="Arg">gens</var> are known to form a basis of the free left module <var class="Arg">V</var>. <code class="func">UseBasis</code> stores information in <var class="Arg">V</var> that can be derived form this fact, namely</p>


<ul>
<li><p><var class="Arg">gens</var> are stored as left module generators if no such generators were bound (this is useful especially if <var class="Arg">V</var> is an algebra),</p>

</li>
<li><p>the dimension of <var class="Arg">V</var> is stored.</p>

</li>
</ul>

<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">V:= FreeLeftModule( Rationals, [ [ 1, 0 ], [ 0, 1 ], [ 1, 1 ] ] );;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">UseBasis( V, [ [ 1, 0 ], [ 1, 1 ] ] );</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">V;  # now V knows its dimension</span>
&lt;vector space of dimension 2 over Rationals&gt;
</pre></div>

<p><a id="X7C8F844783F4FA09" name="X7C8F844783F4FA09"></a></p>

<h5>57.3-6 IsRowModule</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsRowModule</code>( <var class="Arg">V</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>A <em>row module</em> is a free left module whose elements are row vectors.</p>

<p><a id="X81FCC1D780435CF1" name="X81FCC1D780435CF1"></a></p>

<h5>57.3-7 IsMatrixModule</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsMatrixModule</code>( <var class="Arg">V</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>A <em>matrix module</em> is a free left module whose elements are matrices.</p>

<p><a id="X853E085C868196EF" name="X853E085C868196EF"></a></p>

<h5>57.3-8 IsFullRowModule</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsFullRowModule</code>( <var class="Arg">M</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>A <em>full row module</em> is a module <span class="SimpleMath">R^n</span>, for a ring <span class="SimpleMath">R</span> and a nonnegative integer <span class="SimpleMath">n</span>.</p>

<p>More precisely, a full row module is a free left module over a ring <span class="SimpleMath">R</span> such that the elements are row vectors of the same length <span class="SimpleMath">n</span> and with entries in <span class="SimpleMath">R</span> and such that the dimension is equal to <span class="SimpleMath">n</span>.</p>

<p>Several functions delegate their tasks to full row modules, for example <code class="func">Iterator</code> (<a href="chap30.html#X83ADF8287ED0668E"><span class="RefLink">30.8-1</span></a>) and <code class="func">Enumerator</code> (<a href="chap30.html#X7EF8910F82B45EC7"><span class="RefLink">30.3-2</span></a>).</p>

<p><a id="X848041A47BC4B038" name="X848041A47BC4B038"></a></p>

<h5>57.3-9 FullRowModule</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; FullRowModule</code>( <var class="Arg">R</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>is the row module <code class="code"><var class="Arg">R</var>^<var class="Arg">n</var></code>, for a ring <var class="Arg">R</var> and a nonnegative integer <var class="Arg">n</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">V:= FullRowModule( Integers, 5 );</span>
( Integers^5 )
</pre></div>

<p><a id="X814CEA62842CF5BB" name="X814CEA62842CF5BB"></a></p>

<h5>57.3-10 IsFullMatrixModule</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsFullMatrixModule</code>( <var class="Arg">M</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>A <em>full matrix module</em> is a module <span class="SimpleMath">R^{[m,n]}</span>, for a ring <span class="SimpleMath">R</span> and two nonnegative integers <span class="SimpleMath">m</span>, <span class="SimpleMath">n</span>.</p>

<p>More precisely, a full matrix module is a free left module over a ring <span class="SimpleMath">R</span> such that the elements are <span class="SimpleMath">m</span> by <span class="SimpleMath">n</span> matrices with entries in <span class="SimpleMath">R</span> and such that the dimension is equal to <span class="SimpleMath">m n</span>.</p>

<p><a id="X7A0C871B7C446F1F" name="X7A0C871B7C446F1F"></a></p>

<h5>57.3-11 FullMatrixModule</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; FullMatrixModule</code>( <var class="Arg">R</var>, <var class="Arg">m</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>is the matrix module <code class="code"><var class="Arg">R</var>^[<var class="Arg">m</var>,<var class="Arg">n</var>]</code>, for a ring <var class="Arg">R</var> and nonnegative integers <var class="Arg">m</var> and <var class="Arg">n</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">FullMatrixModule( GaussianIntegers, 3, 6 );</span>
( GaussianIntegers^[ 3, 6 ] )
</pre></div>


<div class="chlinkprevnextbot">&nbsp;<a href="chap0.html">[Top of Book]</a>&nbsp;  <a href="chap0.html#contents">[Contents]</a>&nbsp;  &nbsp;<a href="chap56.html">[Previous Chapter]</a>&nbsp;  &nbsp;<a href="chap58.html">[Next Chapter]</a>&nbsp;  </div>


<div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a>  <a href="chap1.html">1</a>  <a href="chap2.html">2</a>  <a href="chap3.html">3</a>  <a href="chap4.html">4</a>  <a href="chap5.html">5</a>  <a href="chap6.html">6</a>  <a href="chap7.html">7</a>  <a href="chap8.html">8</a>  <a href="chap9.html">9</a>  <a href="chap10.html">10</a>  <a href="chap11.html">11</a>  <a href="chap12.html">12</a>  <a href="chap13.html">13</a>  <a href="chap14.html">14</a>  <a href="chap15.html">15</a>  <a href="chap16.html">16</a>  <a href="chap17.html">17</a>  <a href="chap18.html">18</a>  <a href="chap19.html">19</a>  <a href="chap20.html">20</a>  <a href="chap21.html">21</a>  <a href="chap22.html">22</a>  <a href="chap23.html">23</a>  <a href="chap24.html">24</a>  <a href="chap25.html">25</a>  <a href="chap26.html">26</a>  <a href="chap27.html">27</a>  <a href="chap28.html">28</a>  <a href="chap29.html">29</a>  <a href="chap30.html">30</a>  <a href="chap31.html">31</a>  <a href="chap32.html">32</a>  <a href="chap33.html">33</a>  <a href="chap34.html">34</a>  <a href="chap35.html">35</a>  <a href="chap36.html">36</a>  <a href="chap37.html">37</a>  <a href="chap38.html">38</a>  <a href="chap39.html">39</a>  <a href="chap40.html">40</a>  <a href="chap41.html">41</a>  <a href="chap42.html">42</a>  <a href="chap43.html">43</a>  <a href="chap44.html">44</a>  <a href="chap45.html">45</a>  <a href="chap46.html">46</a>  <a href="chap47.html">47</a>  <a href="chap48.html">48</a>  <a href="chap49.html">49</a>  <a href="chap50.html">50</a>  <a href="chap51.html">51</a>  <a href="chap52.html">52</a>  <a href="chap53.html">53</a>  <a href="chap54.html">54</a>  <a href="chap55.html">55</a>  <a href="chap56.html">56</a>  <a href="chap57.html">57</a>  <a href="chap58.html">58</a>  <a href="chap59.html">59</a>  <a href="chap60.html">60</a>  <a href="chap61.html">61</a>  <a href="chap62.html">62</a>  <a href="chap63.html">63</a>  <a href="chap64.html">64</a>  <a href="chap65.html">65</a>  <a href="chap66.html">66</a>  <a href="chap67.html">67</a>  <a href="chap68.html">68</a>  <a href="chap69.html">69</a>  <a href="chap70.html">70</a>  <a href="chap71.html">71</a>  <a href="chap72.html">72</a>  <a href="chap73.html">73</a>  <a href="chap74.html">74</a>  <a href="chap75.html">75</a>  <a href="chap76.html">76</a>  <a href="chap77.html">77</a>  <a href="chap78.html">78</a>  <a href="chap79.html">79</a>  <a href="chap80.html">80</a>  <a href="chap81.html">81</a>  <a href="chap82.html">82</a>  <a href="chap83.html">83</a>  <a href="chap84.html">84</a>  <a href="chap85.html">85</a>  <a href="chap86.html">86</a>  <a href="chap87.html">87</a>  <a href="chapBib.html">Bib</a>  <a href="chapInd.html">Ind</a>  </div>

<hr />
<p class="foot">generated by <a href="http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p>
</body>
</html>