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<div class="ChapSects"><a href="chap61.html#X7DAD6700787EC845">61 <span class="Heading">Vector Spaces</span></a>
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</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap61.html#X79B305CE87511C4B">61.9-1 IsRowSpace</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap61.html#X7A2BBBA07B2BE8F8">61.9-2 IsMatrixSpace</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap61.html#X83724C157F4FDFB4">61.9-3 IsGaussianSpace</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap61.html#X80209A8785126AAB">61.9-4 FullRowSpace</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap61.html#X876B66C37A7B749F">61.9-5 FullMatrixSpace</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap61.html#X8534A750878478D0">61.9-6 DimensionOfVectors</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap61.html#X865A540F85FAE2DF">61.9-7 IsSemiEchelonized</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap61.html#X87DCA09579589106">61.9-8 SemiEchelonBasis</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap61.html#X7C3CC5F97FA048A4">61.9-9 IsCanonicalBasisFullRowModule</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap61.html#X83D282697C1A3148">61.9-10 IsCanonicalBasisFullMatrixModule</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap61.html#X7D6537F87E940344">61.9-11 NormedRowVectors</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap61.html#X815C69A57C042C34">61.9-12 SiftedVector</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap61.html#X7F61CECA84CEF39D">61.10 <span class="Heading">Vector Space Homomorphisms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap61.html#X82013D328645E370">61.10-1 LeftModuleGeneralMappingByImages</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap61.html#X85F5293983E47B5A">61.10-2 LeftModuleHomomorphismByImages</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap61.html#X8477E6C3872A6DBB">61.10-3 LeftModuleHomomorphismByMatrix</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap61.html#X8494AA5D7C3B88AD">61.10-4 NaturalHomomorphismBySubspace</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap61.html#X80015C78876B4F1E">61.10-5 Hom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap61.html#X8680ADD381ECF879">61.10-6 End</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap61.html#X7A9A08EA79259659">61.10-7 IsFullHomModule</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap61.html#X7C4737687E76A24A">61.10-8 IsPseudoCanonicalBasisFullHomModule</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap61.html#X84F87C327A1856F2">61.10-9 IsLinearMappingsModule</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap61.html#X81503EB77FCE648D">61.11 <span class="Heading">Vector Spaces Handled By Nice Bases</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap61.html#X826FD4BC7BA0559D">61.11-1 NiceFreeLeftModule</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap61.html#X807B8032780C59A4">61.11-2 NiceVector</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap61.html#X79350786800C2DD8">61.11-3 NiceFreeLeftModuleInfo</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap61.html#X8388E0248690C214">61.11-4 NiceBasis</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap61.html#X82BC30A487967F5B">61.11-5 IsBasisByNiceBasis</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap61.html#X79D1DEA679AEDA40">61.11-6 IsHandledByNiceBasis</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap61.html#X8238195B851D3C44">61.12 <span class="Heading">How to Implement New Kinds of Vector Spaces</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap61.html#X7DE34C3E837FCBC3">61.12-1 DeclareHandlingByNiceBasis</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap61.html#X7E6077F0830A28DA">61.12-2 NiceBasisFiltersInfo</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap61.html#X7A374553786DF5E7">61.12-3 CheckForHandlingByNiceBasis</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap61.html#X78515F448644204E">61.13 <span class="Heading">Tensor Products and Exterior and Symmetric Powers</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap61.html#X81B2276A7EBA8ED1">61.13-1 TensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap61.html#X787BB7FF85F0AD68">61.13-2 ExteriorPower</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap61.html#X79E2C2AF842E8419">61.13-3 SymmetricPower</a></span>
</div></div>
</div>
<h3>61 <span class="Heading">Vector Spaces</span></h3>
<p><a id="X8754F7207CFDA38B" name="X8754F7207CFDA38B"></a></p>
<h4>61.1 <span class="Heading">IsLeftVectorSpace (Filter)</span></h4>
<p><a id="X80290A908241706B" name="X80290A908241706B"></a></p>
<h5>61.1-1 IsLeftVectorSpace</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsLeftVectorSpace</code>( <var class="Arg">V</var> )</td><td class="tdright">( category )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsVectorSpace</code>( <var class="Arg">V</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>A <em>vector space</em> in <strong class="pkg">GAP</strong> is a free left module (see <code class="func">IsFreeLeftModule</code> (<a href="chap57.html#X7C4832187F3D9228"><span class="RefLink">57.3-1</span></a>)) over a division ring (see Chapter <a href="chap58.html#X80A8E676814A19FD"><span class="RefLink">58</span></a>).</p>
<p>Whenever we talk about an <span class="SimpleMath">F</span>-vector space <var class="Arg">V</var> then <var class="Arg">V</var> is an additive group (see <code class="func">IsAdditiveGroup</code> (<a href="chap55.html#X7B8FBD9082CE271B"><span class="RefLink">55.1-6</span></a>)) on which the division ring <span class="SimpleMath">F</span> acts via multiplication from the left such that this action and the addition in <var class="Arg">V</var> are left and right distributive. The division ring <span class="SimpleMath">F</span> can be accessed as value of the attribute <code class="func">LeftActingDomain</code> (<a href="chap57.html#X86F070E0807DC34E"><span class="RefLink">57.1-11</span></a>).</p>
<p>Vector spaces in <strong class="pkg">GAP</strong> are always <em>left</em> vector spaces, <code class="func">IsLeftVectorSpace</code> and <code class="func">IsVectorSpace</code> are synonyms.</p>
<p><a id="X87AD06FE873619EA" name="X87AD06FE873619EA"></a></p>
<h4>61.2 <span class="Heading">Constructing Vector Spaces</span></h4>
<p><a id="X805413157CE9BECF" name="X805413157CE9BECF"></a></p>
<h5>61.2-1 VectorSpace</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ VectorSpace</code>( <var class="Arg">F</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>For a field <var class="Arg">F</var> and a collection <var class="Arg">gens</var> of vectors, <code class="func">VectorSpace</code> returns the <var class="Arg">F</var>-vector space spanned by the elements in <var class="Arg">gens</var>.</p>
<p>The optional argument <var class="Arg">zero</var> can be used to specify the zero element of the space; <var class="Arg">zero</var> <em>must</em> be given if <var class="Arg">gens</var> is empty. The optional string <code class="code">"basis"</code> indicates that <var class="Arg">gens</var> is known to be linearly independent over <var class="Arg">F</var>, in particular the dimension of the vector space is immediately set; note that <code class="func">Basis</code> (<a href="chap61.html#X837BE54C80DE368E"><span class="RefLink">61.5-2</span></a>) need <em>not</em> return the basis formed by <var class="Arg">gens</var> if the string <code class="code">"basis"</code> is given as an argument.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= VectorSpace( Rationals, [ [ 1, 2, 3 ], [ 1, 1, 1 ] ] );</span>
<vector space over Rationals, with 2 generators>
</pre></div>
<p><a id="X78C9826780BC9AE6" name="X78C9826780BC9AE6"></a></p>
<h5>61.2-2 Subspace</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Subspace</code>( <var class="Arg">V</var>, <var class="Arg">gens</var>[, <var class="Arg">"basis"</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SubspaceNC</code>( <var class="Arg">V</var>, <var class="Arg">gens</var>[, <var class="Arg">"basis"</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>For an <span class="SimpleMath">F</span>-vector space <var class="Arg">V</var> and a list or collection <var class="Arg">gens</var> that is a subset of <var class="Arg">V</var>, <code class="func">Subspace</code> returns the <span class="SimpleMath">F</span>-vector space spanned by <var class="Arg">gens</var>; if <var class="Arg">gens</var> is empty then the trivial subspace (see <code class="func">TrivialSubspace</code> (<a href="chap61.html#X86DC71A9835430FD"><span class="RefLink">61.3-2</span></a>)) of <var class="Arg">V</var> is returned. The parent (see <a href="chap31.html#X7CBDD36E7B7BE286"><span class="RefLink">31.7</span></a>) of the returned vector space is set to <var class="Arg">V</var>.</p>
<p><code class="func">SubspaceNC</code> does the same as <code class="func">Subspace</code>, except that it omits the check whether <var class="Arg">gens</var> is a subset of <var class="Arg">V</var>.</p>
<p>The optional string <var class="Arg">"basis"</var> indicates that <var class="Arg">gens</var> is known to be linearly independent over <span class="SimpleMath">F</span>. In this case the dimension of the subspace is immediately set, and both <code class="func">Subspace</code> and <code class="func">SubspaceNC</code> do <em>not</em> check whether <var class="Arg">gens</var> really is linearly independent and whether <var class="Arg">gens</var> is a subset of <var class="Arg">V</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= VectorSpace( Rationals, [ [ 1, 2, 3 ], [ 1, 1, 1 ] ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">W:= Subspace( V, [ [ 0, 1, 2 ] ] );</span>
<vector space over Rationals, with 1 generator>
</pre></div>
<p><a id="X7B001BAF7D5FD5D0" name="X7B001BAF7D5FD5D0"></a></p>
<h5>61.2-3 AsVectorSpace</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AsVectorSpace</code>( <var class="Arg">F</var>, <var class="Arg">D</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Let <var class="Arg">F</var> be a division ring and <var class="Arg">D</var> a domain. If the elements in <var class="Arg">D</var> form an <var class="Arg">F</var>-vector space then <code class="func">AsVectorSpace</code> returns this <var class="Arg">F</var>-vector space, otherwise <code class="keyw">fail</code> is returned.</p>
<p><code class="func">AsVectorSpace</code> can be used for example to view a given vector space as a vector space over a smaller or larger division ring.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= FullRowSpace( GF( 27 ), 3 );</span>
( GF(3^3)^3 )
<span class="GAPprompt">gap></span> <span class="GAPinput">Dimension( V ); LeftActingDomain( V );</span>
3
GF(3^3)
<span class="GAPprompt">gap></span> <span class="GAPinput">W:= AsVectorSpace( GF( 3 ), V );</span>
<vector space over GF(3), with 9 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">Dimension( W ); LeftActingDomain( W );</span>
9
GF(3)
<span class="GAPprompt">gap></span> <span class="GAPinput">AsVectorSpace( GF( 9 ), V );</span>
fail
</pre></div>
<p><a id="X7D4F84C27EDAC89B" name="X7D4F84C27EDAC89B"></a></p>
<h5>61.2-4 AsSubspace</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AsSubspace</code>( <var class="Arg">V</var>, <var class="Arg">U</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Let <var class="Arg">V</var> be an <span class="SimpleMath">F</span>-vector space, and <var class="Arg">U</var> a collection. If <var class="Arg">U</var> is a subset of <var class="Arg">V</var> such that the elements of <var class="Arg">U</var> form an <span class="SimpleMath">F</span>-vector space then <code class="func">AsSubspace</code> returns this vector space, with parent set to <var class="Arg">V</var> (see <code class="func">AsVectorSpace</code> (<a href="chap61.html#X7B001BAF7D5FD5D0"><span class="RefLink">61.2-3</span></a>)). Otherwise <code class="keyw">fail</code> is returned.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= VectorSpace( Rationals, [ [ 1, 2, 3 ], [ 1, 1, 1 ] ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">W:= VectorSpace( Rationals, [ [ 1/2, 1/2, 1/2 ] ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">U:= AsSubspace( V, W );</span>
<vector space over Rationals, with 1 generator>
<span class="GAPprompt">gap></span> <span class="GAPinput">Parent( U ) = V;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">AsSubspace( V, [ [ 1, 1, 1 ] ] );</span>
fail
</pre></div>
<p><a id="X789FB2D883E53662" name="X789FB2D883E53662"></a></p>
<h4>61.3 <span class="Heading">Operations and Attributes for Vector Spaces</span></h4>
<p><a id="X849651C6830C94A1" name="X849651C6830C94A1"></a></p>
<h5>61.3-1 GeneratorsOfLeftVectorSpace</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneratorsOfLeftVectorSpace</code>( <var class="Arg">V</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneratorsOfVectorSpace</code>( <var class="Arg">V</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>For an <span class="SimpleMath">F</span>-vector space <var class="Arg">V</var>, <code class="func">GeneratorsOfLeftVectorSpace</code> returns a list of vectors in <var class="Arg">V</var> that generate <var class="Arg">V</var> as an <span class="SimpleMath">F</span>-vector space.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfVectorSpace( FullRowSpace( Rationals, 3 ) );</span>
[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ]
</pre></div>
<p><a id="X86DC71A9835430FD" name="X86DC71A9835430FD"></a></p>
<h5>61.3-2 TrivialSubspace</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TrivialSubspace</code>( <var class="Arg">V</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>For a vector space <var class="Arg">V</var>, <code class="func">TrivialSubspace</code> returns the subspace of <var class="Arg">V</var> that consists of the zero vector in <var class="Arg">V</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= GF(3)^3;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">triv:= TrivialSubspace( V );</span>
<vector space of dimension 0 over GF(3)>
<span class="GAPprompt">gap></span> <span class="GAPinput">AsSet( triv );</span>
[ [ 0*Z(3), 0*Z(3), 0*Z(3) ] ]
</pre></div>
<p><a id="X8125675583357131" name="X8125675583357131"></a></p>
<h4>61.4 <span class="Heading">Domains of Subspaces of Vector Spaces</span></h4>
<p><a id="X7975E41A7B29C3FD" name="X7975E41A7B29C3FD"></a></p>
<h5>61.4-1 Subspaces</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Subspaces</code>( <var class="Arg">V</var>[, <var class="Arg">k</var>] )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Called with a finite vector space <var class="Arg">V</var>, <code class="func">Subspaces</code> returns the domain of all subspaces of <var class="Arg">V</var>.</p>
<p>Called with <var class="Arg">V</var> and a nonnegative integer <var class="Arg">k</var>, <code class="func">Subspaces</code> returns the domain of all <var class="Arg">k</var>-dimensional subspaces of <var class="Arg">V</var>.</p>
<p>Special <code class="func">Size</code> (<a href="chap30.html#X858ADA3B7A684421"><span class="RefLink">30.4-6</span></a>) and <code class="func">Iterator</code> (<a href="chap30.html#X83ADF8287ED0668E"><span class="RefLink">30.8-1</span></a>) methods are provided for these domains.</p>
<p><a id="X7A8F5C367FAE3D1B" name="X7A8F5C367FAE3D1B"></a></p>
<h5>61.4-2 IsSubspacesVectorSpace</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSubspacesVectorSpace</code>( <var class="Arg">D</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>The domain of all subspaces of a (finite) vector space or of all subspaces of fixed dimension, as returned by <code class="func">Subspaces</code> (<a href="chap61.html#X7975E41A7B29C3FD"><span class="RefLink">61.4-1</span></a>) (see <code class="func">Subspaces</code> (<a href="chap61.html#X7975E41A7B29C3FD"><span class="RefLink">61.4-1</span></a>)) lies in the category <code class="func">IsSubspacesVectorSpace</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">D:= Subspaces( GF(3)^3 );</span>
Subspaces( ( GF(3)^3 ) )
<span class="GAPprompt">gap></span> <span class="GAPinput">Size( D );</span>
28
<span class="GAPprompt">gap></span> <span class="GAPinput">iter:= Iterator( D );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">NextIterator( iter );</span>
<vector space of dimension 0 over GF(3)>
<span class="GAPprompt">gap></span> <span class="GAPinput">NextIterator( iter );</span>
<vector space of dimension 1 over GF(3)>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSubspacesVectorSpace( D );</span>
true
</pre></div>
<p><a id="X828AA09B87F14FAD" name="X828AA09B87F14FAD"></a></p>
<h4>61.5 <span class="Heading">Bases of Vector Spaces</span></h4>
<p>In <strong class="pkg">GAP</strong>, a <em>basis</em> of a free left <span class="SimpleMath">F</span>-module <span class="SimpleMath">V</span> is a list of vectors <span class="SimpleMath">B = [ v_1, v_2, ..., v_n ]</span> in <span class="SimpleMath">V</span> such that <span class="SimpleMath">V</span> is generated as a left <span class="SimpleMath">F</span>-module by these vectors and such that <span class="SimpleMath">B</span> is linearly independent over <span class="SimpleMath">F</span>. The integer <span class="SimpleMath">n</span> is the dimension of <span class="SimpleMath">V</span> (see <code class="func">Dimension</code> (<a href="chap57.html#X7E6926C6850E7C4E"><span class="RefLink">57.3-3</span></a>)). In particular, as each basis is a list (see Chapter <a href="chap21.html#X7B256AE5780F140A"><span class="RefLink">21</span></a>), it has a length (see <code class="func">Length</code> (<a href="chap21.html#X780769238600AFD1"><span class="RefLink">21.17-5</span></a>)), and the <span class="SimpleMath">i</span>-th vector of <span class="SimpleMath">B</span> can be accessed as <span class="SimpleMath">B[i]</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= Rationals^3;</span>
( Rationals^3 )
<span class="GAPprompt">gap></span> <span class="GAPinput">B:= Basis( V );</span>
CanonicalBasis( ( Rationals^3 ) )
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( B );</span>
3
<span class="GAPprompt">gap></span> <span class="GAPinput">B[1];</span>
[ 1, 0, 0 ]
</pre></div>
<p>The operations described below make sense only for bases of <em>finite</em> dimensional vector spaces. (In practice this means that the vector spaces must be <em>low</em> dimensional, that is, the dimension should not exceed a few hundred.)</p>
<p>Besides the basic operations for lists (see <a href="chap21.html#X7B202D147A5C2884"><span class="RefLink">21.2</span></a>), the <em>basic operations for bases</em> are <code class="func">BasisVectors</code> (<a href="chap61.html#X7B1F17AE8027A590"><span class="RefLink">61.6-1</span></a>), <code class="func">Coefficients</code> (<a href="chap61.html#X80B32F667BF6AFD8"><span class="RefLink">61.6-3</span></a>), <code class="func">LinearCombination</code> (<a href="chap61.html#X7D305AB3834889BF"><span class="RefLink">61.6-4</span></a>), and <code class="func">UnderlyingLeftModule</code> (<a href="chap61.html#X81E8AE88843B70FF"><span class="RefLink">61.6-2</span></a>). These and other operations for arbitrary bases are described in <a href="chap61.html#X839B9C4880EBFB5F"><span class="RefLink">61.6</span></a>.</p>
<p>For special kinds of bases, further operations are defined (see <a href="chap61.html#X82809D6C82DE4EC2"><span class="RefLink">61.7</span></a>).</p>
<p><strong class="pkg">GAP</strong> supports the following three kinds of bases.</p>
<p><em>Relative bases</em> delegate the work to other bases of the same free left module, via basechange matrices (see <code class="func">RelativeBasis</code> (<a href="chap61.html#X8786D40B84120F38"><span class="RefLink">61.5-4</span></a>)).</p>
<p><em>Bases handled by nice bases</em> delegate the work to bases of isomorphic left modules over the same left acting domain (see <a href="chap61.html#X81503EB77FCE648D"><span class="RefLink">61.11</span></a>).</p>
<p>Finally, of course there must be bases in <strong class="pkg">GAP</strong> that really do the work.</p>
<p>For example, in the case of a Gaussian row or matrix space <var class="Arg">V</var> (see <a href="chap61.html#X7D937EBC7DE2819B"><span class="RefLink">61.9</span></a>), <code class="code">Basis( <var class="Arg">V</var> )</code> is a semi-echelonized basis (see <code class="func">IsSemiEchelonized</code> (<a href="chap61.html#X865A540F85FAE2DF"><span class="RefLink">61.9-7</span></a>)) that uses Gaussian elimination; such a basis is of the third kind. <code class="code">Basis( <var class="Arg">V</var>, <var class="Arg">vectors</var> )</code> is either semi-echelonized or a relative basis. Other examples of bases of the third kind are canonical bases of finite fields and of abelian number fields.</p>
<p>Bases handled by nice bases are described in <a href="chap61.html#X81503EB77FCE648D"><span class="RefLink">61.11</span></a>. Examples are non-Gaussian row and matrix spaces, and subspaces of finite fields and abelian number fields that are themselves not fields.</p>
<p><a id="X8739510881F5D862" name="X8739510881F5D862"></a></p>
<h5>61.5-1 IsBasis</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsBasis</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>In <strong class="pkg">GAP</strong>, a <em>basis</em> of a free left module is an object that knows how to compute coefficients w.r.t. its basis vectors (see <code class="func">Coefficients</code> (<a href="chap61.html#X80B32F667BF6AFD8"><span class="RefLink">61.6-3</span></a>)). Bases are constructed by <code class="func">Basis</code> (<a href="chap61.html#X837BE54C80DE368E"><span class="RefLink">61.5-2</span></a>). Each basis is an immutable list, the <span class="SimpleMath">i</span>-th entry being the <span class="SimpleMath">i</span>-th basis vector.</p>
<p>(See <a href="chap61.html#X7C11B9C3819F3EA2"><span class="RefLink">61.8</span></a> for mutable bases.)</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= GF(2)^2;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">B:= Basis( V );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsBasis( B );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsBasis( [ [ 1, 0 ], [ 0, 1 ] ] );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsBasis( Basis( Rationals^2, [ [ 1, 0 ], [ 0, 1 ] ] ) );</span>
true
</pre></div>
<p><a id="X837BE54C80DE368E" name="X837BE54C80DE368E"></a></p>
<h5>61.5-2 Basis</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Basis</code>( <var class="Arg">V</var>[, <var class="Arg">vectors</var>] )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BasisNC</code>( <var class="Arg">V</var>, <var class="Arg">vectors</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Called with a free left <span class="SimpleMath">F</span>-module <var class="Arg">V</var> as the only argument, <code class="func">Basis</code> returns an <span class="SimpleMath">F</span>-basis of <var class="Arg">V</var> whose vectors are not further specified.</p>
<p>If additionally a list <var class="Arg">vectors</var> of vectors in <var class="Arg">V</var> is given that forms an <span class="SimpleMath">F</span>-basis of <var class="Arg">V</var> then <code class="func">Basis</code> returns this basis; if <var class="Arg">vectors</var> is not linearly independent over <span class="SimpleMath">F</span> or does not generate <var class="Arg">V</var> as a free left <span class="SimpleMath">F</span>-module then <code class="keyw">fail</code> is returned.</p>
<p><code class="func">BasisNC</code> does the same as the two argument version of <code class="func">Basis</code>, except that it does not check whether <var class="Arg">vectors</var> form a basis.</p>
<p>If no basis vectors are prescribed then <code class="func">Basis</code> need not compute basis vectors; in this case, the vectors are computed in the first call to <code class="func">BasisVectors</code> (<a href="chap61.html#X7B1F17AE8027A590"><span class="RefLink">61.6-1</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= VectorSpace( Rationals, [ [ 1, 2, 7 ], [ 1/2, 1/3, 5 ] ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">B:= Basis( V );</span>
SemiEchelonBasis( <vector space over Rationals, with
2 generators>, ... )
<span class="GAPprompt">gap></span> <span class="GAPinput">BasisVectors( B );</span>
[ [ 1, 2, 7 ], [ 0, 1, -9/4 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">B:= Basis( V, [ [ 1, 2, 7 ], [ 3, 2, 30 ] ] );</span>
Basis( <vector space over Rationals, with 2 generators>,
[ [ 1, 2, 7 ], [ 3, 2, 30 ] ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">Basis( V, [ [ 1, 2, 3 ] ] );</span>
fail
</pre></div>
<p><a id="X7C8EBFF5805F8C51" name="X7C8EBFF5805F8C51"></a></p>
<h5>61.5-3 CanonicalBasis</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CanonicalBasis</code>( <var class="Arg">V</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>If the vector space <var class="Arg">V</var> supports a <em>canonical basis</em> then <code class="func">CanonicalBasis</code> returns this basis, otherwise <code class="keyw">fail</code> is returned.</p>
<p>The defining property of a canonical basis is that its vectors are uniquely determined by the vector space. If canonical bases exist for two vector spaces over the same left acting domain (see <code class="func">LeftActingDomain</code> (<a href="chap57.html#X86F070E0807DC34E"><span class="RefLink">57.1-11</span></a>)) then the equality of these vector spaces can be decided by comparing the canonical bases.</p>
<p>The exact meaning of a canonical basis depends on the type of <var class="Arg">V</var>. Canonical bases are defined for example for Gaussian row and matrix spaces (see <a href="chap61.html#X7D937EBC7DE2819B"><span class="RefLink">61.9</span></a>).</p>
<p>If one designs a new kind of vector spaces (see <a href="chap61.html#X8238195B851D3C44"><span class="RefLink">61.12</span></a>) and defines a canonical basis for these spaces then the <code class="func">CanonicalBasis</code> method one installs (see <code class="func">InstallMethod</code> (<a href="chap78.html#X837EFDAB7BEF290B"><span class="RefLink">78.3-1</span></a>)) must <em>not</em> call <code class="func">Basis</code> (<a href="chap61.html#X837BE54C80DE368E"><span class="RefLink">61.5-2</span></a>). On the other hand, one probably should install a <code class="func">Basis</code> (<a href="chap61.html#X837BE54C80DE368E"><span class="RefLink">61.5-2</span></a>) method that simply calls <code class="func">CanonicalBasis</code>, the value of the method (see <a href="chap78.html#X795EE8257848B438"><span class="RefLink">78.3</span></a> and <a href="chap78.html#X851FC6387CA2B241"><span class="RefLink">78.4</span></a>) being <code class="code">CANONICAL_BASIS_FLAGS</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">vecs:= [ [ 1, 2, 3 ], [ 1, 1, 1 ], [ 1, 1, 1 ] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= VectorSpace( Rationals, vecs );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">B:= CanonicalBasis( V );</span>
CanonicalBasis( <vector space over Rationals, with 3 generators> )
<span class="GAPprompt">gap></span> <span class="GAPinput">BasisVectors( B );</span>
[ [ 1, 0, -1 ], [ 0, 1, 2 ] ]
</pre></div>
<p><a id="X8786D40B84120F38" name="X8786D40B84120F38"></a></p>
<h5>61.5-4 RelativeBasis</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RelativeBasis</code>( <var class="Arg">B</var>, <var class="Arg">vectors</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RelativeBasisNC</code>( <var class="Arg">B</var>, <var class="Arg">vectors</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>A relative basis is a basis of the free left module <var class="Arg">V</var> that delegates the computation of coefficients etc. to another basis of <var class="Arg">V</var> via a basechange matrix.</p>
<p>Let <var class="Arg">B</var> be a basis of the free left module <var class="Arg">V</var>, and <var class="Arg">vectors</var> a list of vectors in <var class="Arg">V</var>.</p>
<p><code class="func">RelativeBasis</code> checks whether <var class="Arg">vectors</var> form a basis of <var class="Arg">V</var>, and in this case a basis is returned in which <var class="Arg">vectors</var> are the basis vectors; otherwise <code class="keyw">fail</code> is returned.</p>
<p><code class="func">RelativeBasisNC</code> does the same, except that it omits the check.</p>
<p><a id="X839B9C4880EBFB5F" name="X839B9C4880EBFB5F"></a></p>
<h4>61.6 <span class="Heading">Operations for Vector Space Bases</span></h4>
<p><a id="X7B1F17AE8027A590" name="X7B1F17AE8027A590"></a></p>
<h5>61.6-1 BasisVectors</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BasisVectors</code>( <var class="Arg">B</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>For a vector space basis <var class="Arg">B</var>, <code class="code">BasisVectors</code> returns the list of basis vectors of <var class="Arg">B</var>. The lists <var class="Arg">B</var> and <code class="code">BasisVectors( <var class="Arg">B</var> )</code> are equal; the main purpose of <code class="code">BasisVectors</code> is to provide access to a list of vectors that does <em>not</em> know about an underlying vector space.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= VectorSpace( Rationals, [ [ 1, 2, 7 ], [ 1/2, 1/3, 5 ] ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">B:= Basis( V, [ [ 1, 2, 7 ], [ 0, 1, -9/4 ] ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">BasisVectors( B );</span>
[ [ 1, 2, 7 ], [ 0, 1, -9/4 ] ]
</pre></div>
<p><a id="X81E8AE88843B70FF" name="X81E8AE88843B70FF"></a></p>
<h5>61.6-2 UnderlyingLeftModule</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UnderlyingLeftModule</code>( <var class="Arg">B</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>For a basis <var class="Arg">B</var> of a free left module <span class="SimpleMath">V</span>, <code class="func">UnderlyingLeftModule</code> returns <span class="SimpleMath">V</span>.</p>
<p>The reason why a basis stores a free left module is that otherwise one would have to store the basis vectors and the coefficient domain separately. Storing the module allows one for example to deal with bases whose basis vectors have not yet been computed yet (see <code class="func">Basis</code> (<a href="chap61.html#X837BE54C80DE368E"><span class="RefLink">61.5-2</span></a>)); furthermore, in some cases it is convenient to test membership of a vector in the module before computing coefficients w.r.t. a basis.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">B:= Basis( GF(2)^6 );; UnderlyingLeftModule( B );</span>
( GF(2)^6 )
</pre></div>
<p><a id="X80B32F667BF6AFD8" name="X80B32F667BF6AFD8"></a></p>
<h5>61.6-3 Coefficients</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Coefficients</code>( <var class="Arg">B</var>, <var class="Arg">v</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Let <span class="SimpleMath">V</span> be the underlying left module of the basis <var class="Arg">B</var>, and <var class="Arg">v</var> a vector such that the family of <var class="Arg">v</var> is the elements family of the family of <span class="SimpleMath">V</span>. Then <code class="code">Coefficients( <var class="Arg">B</var>, <var class="Arg">v</var> )</code> is the list of coefficients of <var class="Arg">v</var> w.r.t. <var class="Arg">B</var> if <var class="Arg">v</var> lies in <span class="SimpleMath">V</span>, and <code class="keyw">fail</code> otherwise.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= VectorSpace( Rationals, [ [ 1, 2, 7 ], [ 1/2, 1/3, 5 ] ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">B:= Basis( V, [ [ 1, 2, 7 ], [ 0, 1, -9/4 ] ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Coefficients( B, [ 1/2, 1/3, 5 ] );</span>
[ 1/2, -2/3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Coefficients( B, [ 1, 0, 0 ] );</span>
fail
</pre></div>
<p><a id="X7D305AB3834889BF" name="X7D305AB3834889BF"></a></p>
<h5>61.6-4 LinearCombination</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LinearCombination</code>( <var class="Arg">B</var>, <var class="Arg">coeff</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>If <var class="Arg">B</var> is a basis object (see <code class="func">IsBasis</code> (<a href="chap61.html#X8739510881F5D862"><span class="RefLink">61.5-1</span></a>)) or a homogeneous list of length <span class="SimpleMath">n</span>, and <var class="Arg">coeff</var> is a row vector of the same length, <code class="func">LinearCombination</code> returns the vector <span class="SimpleMath">∑_{i = 1}^n <var class="Arg">coeff</var>[i] * <var class="Arg">B</var>[i]</span>.</p>
<p>Perhaps the most important usage is the case where <var class="Arg">B</var> forms a basis.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= VectorSpace( Rationals, [ [ 1, 2, 7 ], [ 1/2, 1/3, 5 ] ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">B:= Basis( V, [ [ 1, 2, 7 ], [ 0, 1, -9/4 ] ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">LinearCombination( B, [ 1/2, -2/3 ] );</span>
[ 1/2, 1/3, 5 ]
</pre></div>
<p><a id="X7EB0D16A7EC2DEE3" name="X7EB0D16A7EC2DEE3"></a></p>
<h5>61.6-5 EnumeratorByBasis</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EnumeratorByBasis</code>( <var class="Arg">B</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>For a basis <var class="Arg">B</var> of the free left <span class="SimpleMath">F</span>-module <span class="SimpleMath">V</span> of dimension <span class="SimpleMath">n</span>, <code class="code">EnumeratorByBasis</code> returns an enumerator that loops over the elements of <span class="SimpleMath">V</span> as linear combinations of the vectors of <var class="Arg">B</var> with coefficients the row vectors in the full row space (see <code class="func">FullRowSpace</code> (<a href="chap61.html#X80209A8785126AAB"><span class="RefLink">61.9-4</span></a>)) of dimension <span class="SimpleMath">n</span> over <span class="SimpleMath">F</span>, in the succession given by the default enumerator of this row space.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= GF(2)^3;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">enum:= EnumeratorByBasis( CanonicalBasis( V ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Print( enum{ [ 1 .. 4 ] }, "\n" );</span>
[ [ 0*Z(2), 0*Z(2), 0*Z(2) ], [ 0*Z(2), 0*Z(2), Z(2)^0 ],
[ 0*Z(2), Z(2)^0, 0*Z(2) ], [ 0*Z(2), Z(2)^0, Z(2)^0 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">B:= Basis( V, [ [ 1, 1, 1 ], [ 1, 1, 0 ], [ 1, 0, 0 ] ] * Z(2) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">enum:= EnumeratorByBasis( B );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Print( enum{ [ 1 .. 4 ] }, "\n" );</span>
[ [ 0*Z(2), 0*Z(2), 0*Z(2) ], [ Z(2)^0, 0*Z(2), 0*Z(2) ],
[ Z(2)^0, Z(2)^0, 0*Z(2) ], [ 0*Z(2), Z(2)^0, 0*Z(2) ] ]
</pre></div>
<p><a id="X855625D47979005D" name="X855625D47979005D"></a></p>
<h5>61.6-6 IteratorByBasis</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IteratorByBasis</code>( <var class="Arg">B</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>For a basis <var class="Arg">B</var> of the free left <span class="SimpleMath">F</span>-module <span class="SimpleMath">V</span> of dimension <span class="SimpleMath">n</span>, <code class="code">IteratorByBasis</code> returns an iterator that loops over the elements of <span class="SimpleMath">V</span> as linear combinations of the vectors of <var class="Arg">B</var> with coefficients the row vectors in the full row space (see <code class="func">FullRowSpace</code> (<a href="chap61.html#X80209A8785126AAB"><span class="RefLink">61.9-4</span></a>)) of dimension <span class="SimpleMath">n</span> over <span class="SimpleMath">F</span>, in the succession given by the default enumerator of this row space.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= GF(2)^3;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">iter:= IteratorByBasis( CanonicalBasis( V ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">for i in [ 1 .. 4 ] do Print( NextIterator( iter ), "\n" ); od;</span>
[ 0*Z(2), 0*Z(2), 0*Z(2) ]
[ 0*Z(2), 0*Z(2), Z(2)^0 ]
[ 0*Z(2), Z(2)^0, 0*Z(2) ]
[ 0*Z(2), Z(2)^0, Z(2)^0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">B:= Basis( V, [ [ 1, 1, 1 ], [ 1, 1, 0 ], [ 1, 0, 0 ] ] * Z(2) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">iter:= IteratorByBasis( B );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">for i in [ 1 .. 4 ] do Print( NextIterator( iter ), "\n" ); od;</span>
[ 0*Z(2), 0*Z(2), 0*Z(2) ]
[ Z(2)^0, 0*Z(2), 0*Z(2) ]
[ Z(2)^0, Z(2)^0, 0*Z(2) ]
[ 0*Z(2), Z(2)^0, 0*Z(2) ]
</pre></div>
<p><a id="X82809D6C82DE4EC2" name="X82809D6C82DE4EC2"></a></p>
<h4>61.7 <span class="Heading">Operations for Special Kinds of Bases</span></h4>
<p><a id="X7CC2B3DD81628CE9" name="X7CC2B3DD81628CE9"></a></p>
<h5>61.7-1 IsCanonicalBasis</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsCanonicalBasis</code>( <var class="Arg">B</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>If the underlying free left module <span class="SimpleMath">V</span> of the basis <var class="Arg">B</var> supports a canonical basis (see <code class="func">CanonicalBasis</code> (<a href="chap61.html#X7C8EBFF5805F8C51"><span class="RefLink">61.5-3</span></a>)) then <code class="func">IsCanonicalBasis</code> returns <code class="keyw">true</code> if <var class="Arg">B</var> is equal to the canonical basis of <span class="SimpleMath">V</span>, and <code class="keyw">false</code> otherwise.</p>
<p><a id="X86DE147F8606B739" name="X86DE147F8606B739"></a></p>
<h5>61.7-2 IsIntegralBasis</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsIntegralBasis</code>( <var class="Arg">B</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Let <var class="Arg">B</var> be an <span class="SimpleMath">S</span>-basis of a <em>field</em> <span class="SimpleMath">F</span> for a subfield <span class="SimpleMath">S</span> of <span class="SimpleMath">F</span>, and let <span class="SimpleMath">R</span> and <span class="SimpleMath">M</span> be the rings of algebraic integers in <span class="SimpleMath">S</span> and <span class="SimpleMath">F</span>, respectively. <code class="code">IsIntegralBasis</code> returns <code class="keyw">true</code> if <var class="Arg">B</var> is also an <span class="SimpleMath">R</span>-basis of <span class="SimpleMath">M</span>, and <code class="keyw">false</code> otherwise.</p>
<p><a id="X7FC051C579D61223" name="X7FC051C579D61223"></a></p>
<h5>61.7-3 IsNormalBasis</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsNormalBasis</code>( <var class="Arg">B</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Let <var class="Arg">B</var> be an <span class="SimpleMath">S</span>-basis of a <em>field</em> <span class="SimpleMath">F</span> for a subfield <span class="SimpleMath">S</span> of <span class="SimpleMath">F</span>. <code class="code">IsNormalBasis</code> returns <code class="keyw">true</code> if <var class="Arg">B</var> is invariant under the Galois group (see <code class="func">GaloisGroup</code> (<a href="chap58.html#X80CAA5BA82F09ED2"><span class="RefLink">58.3-1</span></a>)) of the field extension <span class="SimpleMath">F / S</span>, and <code class="keyw">false</code> otherwise.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">B:= CanonicalBasis( GaussianRationals );</span>
CanonicalBasis( GaussianRationals )
<span class="GAPprompt">gap></span> <span class="GAPinput">IsIntegralBasis( B ); IsNormalBasis( B );</span>
true
false
</pre></div>
<p><a id="X7C11B9C3819F3EA2" name="X7C11B9C3819F3EA2"></a></p>
<h4>61.8 <span class="Heading">Mutable Bases</span></h4>
<p>It is useful to have a <em>mutable basis</em> of a free module when successively closures with new vectors are formed, since one does not want to create a new module and a corresponding basis for each step.</p>
<p>Note that the situation here is different from the situation with stabilizer chains, which are (mutable or immutable) records that do not need to know about the groups they describe, whereas each (immutable) basis stores the underlying left module (see <code class="func">UnderlyingLeftModule</code> (<a href="chap61.html#X81E8AE88843B70FF"><span class="RefLink">61.6-2</span></a>)).</p>
<p>So immutable bases and mutable bases are different categories of objects. The only thing they have in common is that one can ask both for their basis vectors and for the coefficients of a given vector.</p>
<p>Since <code class="code">Immutable</code> produces an immutable copy of any <strong class="pkg">GAP</strong> object, it would in principle be possible to construct a mutable basis that is in fact immutable. In the sequel, we will deal only with mutable bases that are in fact <em>mutable</em> <strong class="pkg">GAP</strong> objects, hence these objects are unable to store attribute values.</p>
<p>Basic operations for immutable bases are <code class="func">NrBasisVectors</code> (<a href="chap61.html#X7EC90F4F7BCAF8D4"><span class="RefLink">61.8-3</span></a>), <code class="func">IsContainedInSpan</code> (<a href="chap61.html#X85B50AC77A22108B"><span class="RefLink">61.8-5</span></a>), <code class="func">CloseMutableBasis</code> (<a href="chap61.html#X7B52C99B84316F61"><span class="RefLink">61.8-6</span></a>), <code class="func">ImmutableBasis</code> (<a href="chap61.html#X7BA87512823A8CFD"><span class="RefLink">61.8-4</span></a>), <code class="func">Coefficients</code> (<a href="chap61.html#X80B32F667BF6AFD8"><span class="RefLink">61.6-3</span></a>), and <code class="func">BasisVectors</code> (<a href="chap61.html#X7B1F17AE8027A590"><span class="RefLink">61.6-1</span></a>). <code class="func">ShallowCopy</code> (<a href="chap12.html#X846BC7107C352031"><span class="RefLink">12.7-1</span></a>) for a mutable basis returns a mutable plain list containing the current basis vectors.</p>
<p>Since mutable bases do not admit arbitrary changes of their lists of basis vectors, a mutable basis is <em>not</em> a list. It is, however, a collection, more precisely its family (see <a href="chap13.html#X846063757EC05986"><span class="RefLink">13.1</span></a>) equals the family of its collection of basis vectors.</p>
<p>Mutable bases can be constructed with <code class="code">MutableBasis</code>.</p>
<p>Similar to the situation with bases (cf. <a href="chap61.html#X828AA09B87F14FAD"><span class="RefLink">61.5</span></a>), <strong class="pkg">GAP</strong> supports the following three kinds of mutable bases.</p>
<p>The <em>generic method</em> of <code class="code">MutableBasis</code> returns a mutable basis that simply stores an immutable basis; clearly one wants to avoid this whenever possible with reasonable effort.</p>
<p>There are mutable bases that store a mutable basis for a nicer module. Note that this is meaningful only if the mechanism of computing nice and ugly vectors (see <a href="chap61.html#X81503EB77FCE648D"><span class="RefLink">61.11</span></a>) is invariant under closures of the basis; this is the case for example if the vectors are matrices, Lie objects, or elements of structure constants algebras.</p>
<p>There are mutable bases that use special information to perform their tasks; examples are mutable bases of Gaussian row and matrix spaces.</p>
<p><a id="X7F466FB47F7E9F00" name="X7F466FB47F7E9F00"></a></p>
<h5>61.8-1 IsMutableBasis</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsMutableBasis</code>( <var class="Arg">MB</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>Every mutable basis lies in the category <code class="code">IsMutableBasis</code>.</p>
<p><a id="X8115C061819E5172" name="X8115C061819E5172"></a></p>
<h5>61.8-2 MutableBasis</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MutableBasis</code>( <var class="Arg">R</var>, <var class="Arg">vectors</var>[, <var class="Arg">zero</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p><code class="code">MutableBasis</code> returns a mutable basis for the <var class="Arg">R</var>-free module generated by the vectors in the list <var class="Arg">vectors</var>. The optional argument <var class="Arg">zero</var> is the zero vector of the module; it must be given if <var class="Arg">vectors</var> is empty.</p>
<p><em>Note</em> that <var class="Arg">vectors</var> will in general <em>not</em> be the basis vectors of the mutable basis!</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">MB:= MutableBasis( Rationals, [ [ 1, 2, 3 ], [ 0, 1, 0 ] ] );</span>
<mutable basis over Rationals, 2 vectors>
</pre></div>
<p><a id="X7EC90F4F7BCAF8D4" name="X7EC90F4F7BCAF8D4"></a></p>
<h5>61.8-3 NrBasisVectors</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NrBasisVectors</code>( <var class="Arg">MB</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>For a mutable basis <var class="Arg">MB</var>, <code class="code">NrBasisVectors</code> returns the current number of basis vectors of <var class="Arg">MB</var>. Note that this operation is <em>not</em> an attribute, as it makes no sense to store the value. <code class="code">NrBasisVectors</code> is used mainly as an equivalent of <code class="code">Dimension</code> for the underlying left module in the case of immutable bases.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">MB:= MutableBasis( Rationals, [ [ 1, 1], [ 2, 2 ] ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">NrBasisVectors( MB );</span>
1
</pre></div>
<p><a id="X7BA87512823A8CFD" name="X7BA87512823A8CFD"></a></p>
<h5>61.8-4 ImmutableBasis</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ImmutableBasis</code>( <var class="Arg">MB</var>[, <var class="Arg">V</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p><code class="func">ImmutableBasis</code> returns the immutable basis <span class="SimpleMath">B</span> with the same basis vectors as in the mutable basis <var class="Arg">MB</var>.</p>
<p>If the second argument <var class="Arg">V</var> is present then <var class="Arg">V</var> is the value of <code class="func">UnderlyingLeftModule</code> (<a href="chap61.html#X81E8AE88843B70FF"><span class="RefLink">61.6-2</span></a>) for <span class="SimpleMath">B</span>. The second variant is used mainly for the case that one knows the module for the desired basis in advance, and if it has a nicer structure than the module known to <var class="Arg">MB</var>, for example if it is an algebra.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">MB:= MutableBasis( Rationals, [ [ 1, 1 ], [ 2, 2 ] ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">B:= ImmutableBasis( MB );</span>
SemiEchelonBasis( <vector space of dimension 1 over Rationals>,
[ [ 1, 1 ] ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">UnderlyingLeftModule( B );</span>
<vector space of dimension 1 over Rationals>
</pre></div>
<p><a id="X85B50AC77A22108B" name="X85B50AC77A22108B"></a></p>
<h5>61.8-5 IsContainedInSpan</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsContainedInSpan</code>( <var class="Arg">MB</var>, <var class="Arg">v</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>For a mutable basis <var class="Arg">MB</var> over the coefficient ring <span class="SimpleMath">R</span> and a vector <var class="Arg">v</var>, <code class="code">IsContainedInSpan</code> returns <code class="keyw">true</code> is <var class="Arg">v</var> lies in the <span class="SimpleMath">R</span>-span of the current basis vectors of <var class="Arg">MB</var>, and <code class="keyw">false</code> otherwise.</p>
<p><a id="X7B52C99B84316F61" name="X7B52C99B84316F61"></a></p>
<h5>61.8-6 CloseMutableBasis</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CloseMutableBasis</code>( <var class="Arg">MB</var>, <var class="Arg">v</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>For a mutable basis <var class="Arg">MB</var> over the coefficient ring <span class="SimpleMath">R</span> and a vector <var class="Arg">v</var>, <code class="func">CloseMutableBasis</code> changes <var class="Arg">MB</var> such that afterwards it describes the <span class="SimpleMath">R</span>-span of the former basis vectors together with <var class="Arg">v</var>.</p>
<p><em>Note</em> that if <var class="Arg">v</var> enlarges the dimension then this does in general <em>not</em> mean that <var class="Arg">v</var> is simply added to the basis vectors of <var class="Arg">MB</var>. Usually a linear combination of <var class="Arg">v</var> and the other basis vectors is added, and also the old basis vectors may be modified, for example in order to keep the list of basis vectors echelonized (see <code class="func">IsSemiEchelonized</code> (<a href="chap61.html#X865A540F85FAE2DF"><span class="RefLink">61.9-7</span></a>)).</p>
<p><code class="func">CloseMutableBasis</code> returns <code class="keyw">false</code> if <var class="Arg">v</var> was already in the <span class="SimpleMath">R</span>-span described by <var class="Arg">MB</var>, and <code class="keyw">true</code> if <var class="Arg">MB</var> got extended.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">MB:= MutableBasis( Rationals, [ [ 1, 1, 3 ], [ 2, 2, 1 ] ] );</span>
<mutable basis over Rationals, 2 vectors>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsContainedInSpan( MB, [ 1, 0, 0 ] );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">CloseMutableBasis( MB, [ 1, 0, 0 ] );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">MB;</span>
<mutable basis over Rationals, 3 vectors>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsContainedInSpan( MB, [ 1, 0, 0 ] );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">CloseMutableBasis( MB, [ 1, 0, 0 ] );</span>
false
</pre></div>
<p><a id="X7D937EBC7DE2819B" name="X7D937EBC7DE2819B"></a></p>
<h4>61.9 <span class="Heading">Row and Matrix Spaces</span></h4>
<p><a id="X79B305CE87511C4B" name="X79B305CE87511C4B"></a></p>
<h5>61.9-1 IsRowSpace</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsRowSpace</code>( <var class="Arg">V</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<p>A <em>row space</em> in <strong class="pkg">GAP</strong> is a vector space that consists of row vectors (see Chapter <a href="chap23.html#X82C7E6CF7BA03391"><span class="RefLink">23</span></a>).</p>
<p><a id="X7A2BBBA07B2BE8F8" name="X7A2BBBA07B2BE8F8"></a></p>
<h5>61.9-2 IsMatrixSpace</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsMatrixSpace</code>( <var class="Arg">V</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<p>A <em>matrix space</em> in <strong class="pkg">GAP</strong> is a vector space that consists of matrices (see Chapter <a href="chap24.html#X812CCAB278643A59"><span class="RefLink">24</span></a>).</p>
<p><a id="X83724C157F4FDFB4" name="X83724C157F4FDFB4"></a></p>
<h5>61.9-3 IsGaussianSpace</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGaussianSpace</code>( <var class="Arg">V</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<p>The filter <code class="func">IsGaussianSpace</code> (see <a href="chap13.html#X84EFA4C07D4277BB"><span class="RefLink">13.2</span></a>) for the row space (see <code class="func">IsRowSpace</code> (<a href="chap61.html#X79B305CE87511C4B"><span class="RefLink">61.9-1</span></a>)) or matrix space (see <code class="func">IsMatrixSpace</code> (<a href="chap61.html#X7A2BBBA07B2BE8F8"><span class="RefLink">61.9-2</span></a>)) <var class="Arg">V</var> over a field <span class="SimpleMath">F</span> indicates that the entries of all row vectors or matrices in <var class="Arg">V</var>, respectively, are all contained in <span class="SimpleMath">F</span>. In this case, <var class="Arg">V</var> is called a <em>Gaussian</em> vector space. Bases for Gaussian spaces can be computed using Gaussian elimination for a given list of vector space generators.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">mats:= [ [[1,1],[2,2]], [[3,4],[0,1]] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= VectorSpace( Rationals, mats );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsGaussianSpace( V );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">mats[1][1][1]:= E(4);; # an element in an extension field</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= VectorSpace( Rationals, mats );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsGaussianSpace( V );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= VectorSpace( Field( Rationals, [ E(4) ] ), mats );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsGaussianSpace( V );</span>
true
</pre></div>
<p><a id="X80209A8785126AAB" name="X80209A8785126AAB"></a></p>
<h5>61.9-4 FullRowSpace</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FullRowSpace</code>( <var class="Arg">F</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ \^</code>( <var class="Arg">F</var>, <var class="Arg">n</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>For a field <var class="Arg">F</var> and a nonnegative integer <var class="Arg">n</var>, <code class="func">FullRowSpace</code> returns the <var class="Arg">F</var>-vector space that consists of all row vectors (see <code class="func">IsRowVector</code> (<a href="chap23.html#X7DFB22A07836A7A9"><span class="RefLink">23.1-1</span></a>)) of length <var class="Arg">n</var> with entries in <var class="Arg">F</var>.</p>
<p>An alternative to construct this vector space is via <var class="Arg">F</var><code class="code">^</code><var class="Arg">n</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">FullRowSpace( GF( 9 ), 3 );</span>
( GF(3^2)^3 )
<span class="GAPprompt">gap></span> <span class="GAPinput">GF(9)^3; # the same as above</span>
( GF(3^2)^3 )
</pre></div>
<p><a id="X876B66C37A7B749F" name="X876B66C37A7B749F"></a></p>
<h5>61.9-5 FullMatrixSpace</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FullMatrixSpace</code>( <var class="Arg">F</var>, <var class="Arg">m</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ \^</code>( <var class="Arg">F</var>, <var class="Arg">dims</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>For a field <var class="Arg">F</var> and two positive integers <var class="Arg">m</var> and <var class="Arg">n</var>, <code class="func">FullMatrixSpace</code> returns the <var class="Arg">F</var>-vector space that consists of all <var class="Arg">m</var> by <var class="Arg">n</var> matrices (see <code class="func">IsMatrix</code> (<a href="chap24.html#X7E1AE46B862B185F"><span class="RefLink">24.2-1</span></a>)) with entries in <var class="Arg">F</var>.</p>
<p>If <var class="Arg">m</var><code class="code"> = </code><var class="Arg">n</var> then the result is in fact an algebra (see <code class="func">FullMatrixAlgebra</code> (<a href="chap62.html#X7D88E42B7DE087B0"><span class="RefLink">62.5-4</span></a>)).</p>
<p>An alternative to construct this vector space is via <var class="Arg">F</var><code class="code">^[</code><var class="Arg">m</var>,<var class="Arg">n</var><code class="code">]</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">FullMatrixSpace( GF(2), 4, 5 );</span>
( GF(2)^[ 4, 5 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">GF(2)^[ 4, 5 ]; # the same as above</span>
( GF(2)^[ 4, 5 ] )
</pre></div>
<p><a id="X8534A750878478D0" name="X8534A750878478D0"></a></p>
<h5>61.9-6 DimensionOfVectors</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DimensionOfVectors</code>( <var class="Arg">M</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>For a left module <var class="Arg">M</var> that consists of row vectors (see <code class="func">IsRowModule</code> (<a href="chap57.html#X7C8F844783F4FA09"><span class="RefLink">57.3-6</span></a>)), <code class="func">DimensionOfVectors</code> returns the common length of all row vectors in <var class="Arg">M</var>. For a left module <var class="Arg">M</var> that consists of matrices (see <code class="func">IsMatrixModule</code> (<a href="chap57.html#X81FCC1D780435CF1"><span class="RefLink">57.3-7</span></a>)), <code class="func">DimensionOfVectors</code> returns the common matrix dimensions (see <code class="func">DimensionsMat</code> (<a href="chap24.html#X83A9DC2085D3A972"><span class="RefLink">24.4-1</span></a>)) of all matrices in <var class="Arg">M</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">DimensionOfVectors( GF(2)^5 );</span>
5
<span class="GAPprompt">gap></span> <span class="GAPinput">DimensionOfVectors( GF(2)^[2,3] );</span>
[ 2, 3 ]
</pre></div>
<p><a id="X865A540F85FAE2DF" name="X865A540F85FAE2DF"></a></p>
<h5>61.9-7 IsSemiEchelonized</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSemiEchelonized</code>( <var class="Arg">B</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Let <var class="Arg">B</var> be a basis of a Gaussian row or matrix space <span class="SimpleMath">V</span> (see <code class="func">IsGaussianSpace</code> (<a href="chap61.html#X83724C157F4FDFB4"><span class="RefLink">61.9-3</span></a>)) over the field <span class="SimpleMath">F</span>.</p>
<p>If <span class="SimpleMath">V</span> is a row space then <var class="Arg">B</var> is semi-echelonized if the matrix formed by its basis vectors has the property that the first nonzero element in each row is the identity of <span class="SimpleMath">F</span>, and all values exactly below these pivot elements are the zero of <span class="SimpleMath">F</span> (cf. <code class="func">SemiEchelonMat</code> (<a href="chap24.html#X7D5D6BD07B7E981B"><span class="RefLink">24.10-1</span></a>)).</p>
<p>If <span class="SimpleMath">V</span> is a matrix space then <var class="Arg">B</var> is semi-echelonized if the matrix obtained by replacing each basis vector by the concatenation of its rows is semi-echelonized (see above, cf. <code class="func">SemiEchelonMats</code> (<a href="chap24.html#X827D7971800DB661"><span class="RefLink">24.10-4</span></a>)).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= GF(2)^2;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">B1:= Basis( V, [ [ 0, 1 ], [ 1, 0 ] ] * Z(2) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSemiEchelonized( B1 );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">B2:= Basis( V, [ [ 0, 1 ], [ 1, 1 ] ] * Z(2) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSemiEchelonized( B2 );</span>
false
</pre></div>
<p><a id="X87DCA09579589106" name="X87DCA09579589106"></a></p>
<h5>61.9-8 SemiEchelonBasis</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SemiEchelonBasis</code>( <var class="Arg">V</var>[, <var class="Arg">vectors</var>] )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SemiEchelonBasisNC</code>( <var class="Arg">V</var>, <var class="Arg">vectors</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Let <var class="Arg">V</var> be a Gaussian row or matrix vector space over the field <span class="SimpleMath">F</span> (see <code class="func">IsGaussianSpace</code> (<a href="chap61.html#X83724C157F4FDFB4"><span class="RefLink">61.9-3</span></a>), <code class="func">IsRowSpace</code> (<a href="chap61.html#X79B305CE87511C4B"><span class="RefLink">61.9-1</span></a>), <code class="func">IsMatrixSpace</code> (<a href="chap61.html#X7A2BBBA07B2BE8F8"><span class="RefLink">61.9-2</span></a>)).</p>
<p>Called with <var class="Arg">V</var> as the only argument, <code class="func">SemiEchelonBasis</code> returns a basis of <var class="Arg">V</var> that has the property <code class="func">IsSemiEchelonized</code> (<a href="chap61.html#X865A540F85FAE2DF"><span class="RefLink">61.9-7</span></a>).</p>
<p>If additionally a list <var class="Arg">vectors</var> of vectors in <var class="Arg">V</var> is given that forms a semi-echelonized basis of <var class="Arg">V</var> then <code class="func">SemiEchelonBasis</code> returns this basis; if <var class="Arg">vectors</var> do not form a basis of <var class="Arg">V</var> then <code class="keyw">fail</code> is returned.</p>
<p><code class="func">SemiEchelonBasisNC</code> does the same as the two argument version of <code class="func">SemiEchelonBasis</code>, except that it is not checked whether <var class="Arg">vectors</var> form a semi-echelonized basis.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= GF(2)^2;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">B:= SemiEchelonBasis( V );</span>
SemiEchelonBasis( ( GF(2)^2 ), ... )
<span class="GAPprompt">gap></span> <span class="GAPinput">Print( BasisVectors( B ), "\n" );</span>
[ [ Z(2)^0, 0*Z(2) ], [ 0*Z(2), Z(2)^0 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">B:= SemiEchelonBasis( V, [ [ 1, 1 ], [ 0, 1 ] ] * Z(2) );</span>
SemiEchelonBasis( ( GF(2)^2 ), <an immutable 2x2 matrix over GF2> )
<span class="GAPprompt">gap></span> <span class="GAPinput">Print( BasisVectors( B ), "\n" );</span>
[ [ Z(2)^0, Z(2)^0 ], [ 0*Z(2), Z(2)^0 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Coefficients( B, [ 0, 1 ] * Z(2) );</span>
[ 0*Z(2), Z(2)^0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Coefficients( B, [ 1, 0 ] * Z(2) );</span>
[ Z(2)^0, Z(2)^0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">SemiEchelonBasis( V, [ [ 0, 1 ], [ 1, 1 ] ] * Z(2) );</span>
fail
</pre></div>
<p><a id="X7C3CC5F97FA048A4" name="X7C3CC5F97FA048A4"></a></p>
<h5>61.9-9 IsCanonicalBasisFullRowModule</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsCanonicalBasisFullRowModule</code>( <var class="Arg">B</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p><code class="func">IsCanonicalBasisFullRowModule</code> returns <code class="keyw">true</code> if <var class="Arg">B</var> is the canonical basis (see <code class="func">IsCanonicalBasis</code> (<a href="chap61.html#X7CC2B3DD81628CE9"><span class="RefLink">61.7-1</span></a>)) of a full row module (see <code class="func">IsFullRowModule</code> (<a href="chap57.html#X853E085C868196EF"><span class="RefLink">57.3-8</span></a>)), and <code class="keyw">false</code> otherwise.</p>
<p>The <em>canonical basis</em> of a Gaussian row space is defined as the unique semi-echelonized (see <code class="func">IsSemiEchelonized</code> (<a href="chap61.html#X865A540F85FAE2DF"><span class="RefLink">61.9-7</span></a>)) basis with the additional property that for <span class="SimpleMath">j > i</span> the position of the pivot of row <span class="SimpleMath">j</span> is bigger than the position of the pivot of row <span class="SimpleMath">i</span>, and that each pivot column contains exactly one nonzero entry.</p>
<p><a id="X83D282697C1A3148" name="X83D282697C1A3148"></a></p>
<h5>61.9-10 IsCanonicalBasisFullMatrixModule</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsCanonicalBasisFullMatrixModule</code>( <var class="Arg">B</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p><code class="func">IsCanonicalBasisFullMatrixModule</code> returns <code class="keyw">true</code> if <var class="Arg">B</var> is the canonical basis (see <code class="func">IsCanonicalBasis</code> (<a href="chap61.html#X7CC2B3DD81628CE9"><span class="RefLink">61.7-1</span></a>)) of a full matrix module (see <code class="func">IsFullMatrixModule</code> (<a href="chap57.html#X814CEA62842CF5BB"><span class="RefLink">57.3-10</span></a>)), and <code class="keyw">false</code> otherwise.</p>
<p>The <em>canonical basis</em> of a Gaussian matrix space is defined as the unique semi-echelonized (see <code class="func">IsSemiEchelonized</code> (<a href="chap61.html#X865A540F85FAE2DF"><span class="RefLink">61.9-7</span></a>)) basis for which the list of concatenations of the basis vectors forms the canonical basis of the corresponding Gaussian row space.</p>
<p><a id="X7D6537F87E940344" name="X7D6537F87E940344"></a></p>
<h5>61.9-11 NormedRowVectors</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NormedRowVectors</code>( <var class="Arg">V</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>For a finite Gaussian row space <var class="Arg">V</var> (see <code class="func">IsRowSpace</code> (<a href="chap61.html#X79B305CE87511C4B"><span class="RefLink">61.9-1</span></a>), <code class="func">IsGaussianSpace</code> (<a href="chap61.html#X83724C157F4FDFB4"><span class="RefLink">61.9-3</span></a>)), <code class="func">NormedRowVectors</code> returns a list of those nonzero vectors in <var class="Arg">V</var> that have a one in the first nonzero component.</p>
<p>The result list can be used as action domain for the action of a matrix group via <code class="func">OnLines</code> (<a href="chap41.html#X86DC2DD5829CAD9A"><span class="RefLink">41.2-12</span></a>), which yields the natural action on one-dimensional subspaces of <var class="Arg">V</var> (see also <code class="func">Subspaces</code> (<a href="chap61.html#X7975E41A7B29C3FD"><span class="RefLink">61.4-1</span></a>)).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">vecs:= NormedRowVectors( GF(3)^2 );</span>
[ [ 0*Z(3), Z(3)^0 ], [ Z(3)^0, 0*Z(3) ], [ Z(3)^0, Z(3)^0 ],
[ Z(3)^0, Z(3) ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Action( GL(2,3), vecs, OnLines );</span>
Group([ (3,4), (1,2,4) ])
</pre></div>
<p><a id="X815C69A57C042C34" name="X815C69A57C042C34"></a></p>
<h5>61.9-12 SiftedVector</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SiftedVector</code>( <var class="Arg">B</var>, <var class="Arg">v</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Let <var class="Arg">B</var> be a semi-echelonized basis (see <code class="func">IsSemiEchelonized</code> (<a href="chap61.html#X865A540F85FAE2DF"><span class="RefLink">61.9-7</span></a>)) of a Gaussian row or matrix space <span class="SimpleMath">V</span> (see <code class="func">IsGaussianSpace</code> (<a href="chap61.html#X83724C157F4FDFB4"><span class="RefLink">61.9-3</span></a>)), and <var class="Arg">v</var> a row vector or matrix, respectively, of the same dimension as the elements in <span class="SimpleMath">V</span>. <code class="code">SiftedVector</code> returns the <em>residuum</em> of <var class="Arg">v</var> with respect to <var class="Arg">B</var>, which is obtained by successively cleaning the pivot positions in <var class="Arg">v</var> by subtracting multiples of the basis vectors in <var class="Arg">B</var>. So the result is the zero vector in <span class="SimpleMath">V</span> if and only if <var class="Arg">v</var> lies in <span class="SimpleMath">V</span>.</p>
<p><var class="Arg">B</var> may also be a mutable basis (see <a href="chap61.html#X7C11B9C3819F3EA2"><span class="RefLink">61.8</span></a>) of a Gaussian row or matrix space.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= VectorSpace( Rationals, [ [ 1, 2, 7 ], [ 1/2, 1/3, 5 ] ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">B:= Basis( V );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SiftedVector( B, [ 1, 2, 8 ] );</span>
[ 0, 0, 1 ]
</pre></div>
<p><a id="X7F61CECA84CEF39D" name="X7F61CECA84CEF39D"></a></p>
<h4>61.10 <span class="Heading">Vector Space Homomorphisms</span></h4>
<p><em>Vector space homomorphisms</em> (or <em>linear mappings</em>) are defined in Section <a href="chap32.html#X7C24431C81532575"><span class="RefLink">32.11</span></a>. <strong class="pkg">GAP</strong> 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>
<p><span class="SimpleMath">F</span>-linear mappings with same source and same range can be added, so one can form vector spaces of linear mappings.</p>
<p><a id="X82013D328645E370" name="X82013D328645E370"></a></p>
<h5>61.10-1 LeftModuleGeneralMappingByImages</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftModuleGeneralMappingByImages</code>( <var class="Arg">V</var>, <var class="Arg">W</var>, <var class="Arg">gens</var>, <var class="Arg">imgs</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Let <var class="Arg">V</var> and <var class="Arg">W</var> be two left modules over the same left acting domain <span class="SimpleMath">R</span> and <var class="Arg">gens</var> and <var class="Arg">imgs</var> lists (of the same length) of elements in <var class="Arg">V</var> and <var class="Arg">W</var>, respectively. <code class="func">LeftModuleGeneralMappingByImages</code> returns the general mapping with source <var class="Arg">V</var> and range <var class="Arg">W</var> that is defined by mapping the elements in <var class="Arg">gens</var> to the corresponding elements in <var class="Arg">imgs</var>, and taking the <span class="SimpleMath">R</span>-linear closure.</p>
<p><var class="Arg">gens</var> need not generate <var class="Arg">V</var> as a left <span class="SimpleMath">R</span>-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 <code class="func">IsMapping</code> (<a href="chap32.html#X7CC95EB282854385"><span class="RefLink">32.3-3</span></a>)).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= Rationals^2;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">W:= VectorSpace( Rationals, [ [1,2,3], [1,0,1] ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">f:= LeftModuleGeneralMappingByImages( V, W,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [[1,0],[2,0]], [[1,0,1],[1,0,1] ] );</span>
[ [ 1, 0 ], [ 2, 0 ] ] -> [ [ 1, 0, 1 ], [ 1, 0, 1 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsMapping( f );</span>
false
</pre></div>
<p><a id="X85F5293983E47B5A" name="X85F5293983E47B5A"></a></p>
<h5>61.10-2 LeftModuleHomomorphismByImages</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftModuleHomomorphismByImages</code>( <var class="Arg">V</var>, <var class="Arg">W</var>, <var class="Arg">gens</var>, <var class="Arg">imgs</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftModuleHomomorphismByImagesNC</code>( <var class="Arg">V</var>, <var class="Arg">W</var>, <var class="Arg">gens</var>, <var class="Arg">imgs</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Let <var class="Arg">V</var> and <var class="Arg">W</var> be two left modules over the same left acting domain <span class="SimpleMath">R</span> and <var class="Arg">gens</var> and <var class="Arg">imgs</var> lists (of the same length) of elements in <var class="Arg">V</var> and <var class="Arg">W</var>, respectively. <code class="func">LeftModuleHomomorphismByImages</code> returns the left <span class="SimpleMath">R</span>-module homomorphism with source <var class="Arg">V</var> and range <var class="Arg">W</var> that is defined by mapping the elements in <var class="Arg">gens</var> to the corresponding elements in <var class="Arg">imgs</var>.</p>
<p>If <var class="Arg">gens</var> does not generate <var class="Arg">V</var> or if the homomorphism does not exist (i.e., if mapping the generators describes only a multi-valued mapping) then <code class="keyw">fail</code> is returned. For creating a possibly multi-valued mapping from <var class="Arg">V</var> to <var class="Arg">W</var> that respects addition, multiplication, and scalar multiplication, <code class="func">LeftModuleGeneralMappingByImages</code> (<a href="chap61.html#X82013D328645E370"><span class="RefLink">61.10-1</span></a>) can be used.</p>
<p><code class="func">LeftModuleHomomorphismByImagesNC</code> does the same as <code class="func">LeftModuleHomomorphismByImages</code>, except that it omits all checks.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:=Rationals^2;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">W:=VectorSpace( Rationals, [ [ 1, 0, 1 ], [ 1, 2, 3 ] ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">f:=LeftModuleHomomorphismByImages( V, W,</span>
<span class="GAPprompt">></span> <span class="GAPinput">[ [ 1, 0 ], [ 0, 1 ] ], [ [ 1, 0, 1 ], [ 1, 2, 3 ] ] );</span>
[ [ 1, 0 ], [ 0, 1 ] ] -> [ [ 1, 0, 1 ], [ 1, 2, 3 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Image( f, [1,1] );</span>
[ 2, 2, 4 ]
</pre></div>
<p><a id="X8477E6C3872A6DBB" name="X8477E6C3872A6DBB"></a></p>
<h5>61.10-3 LeftModuleHomomorphismByMatrix</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftModuleHomomorphismByMatrix</code>( <var class="Arg">BS</var>, <var class="Arg">matrix</var>, <var class="Arg">BR</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Let <var class="Arg">BS</var> and <var class="Arg">BR</var> be bases of the left <span class="SimpleMath">R</span>-modules <span class="SimpleMath">V</span> and <span class="SimpleMath">W</span>, respectively. <code class="func">LeftModuleHomomorphismByMatrix</code> returns the <span class="SimpleMath">R</span>-linear mapping from <span class="SimpleMath">V</span> to <span class="SimpleMath">W</span> that is defined by the matrix <var class="Arg">matrix</var>, as follows. The image of the <span class="SimpleMath">i</span>-th basis vector of <var class="Arg">BS</var> is the linear combination of the basis vectors of <var class="Arg">BR</var> with coefficients the <span class="SimpleMath">i</span>-th row of <var class="Arg">matrix</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= Rationals^2;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">W:= VectorSpace( Rationals, [ [ 1, 0, 1 ], [ 1, 2, 3 ] ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">f:= LeftModuleHomomorphismByMatrix( Basis( V ),</span>
<span class="GAPprompt">></span> <span class="GAPinput">[ [ 1, 2 ], [ 3, 1 ] ], Basis( W ) );</span>
<linear mapping by matrix, ( Rationals^
2 ) -> <vector space over Rationals, with 2 generators>>
</pre></div>
<p><a id="X8494AA5D7C3B88AD" name="X8494AA5D7C3B88AD"></a></p>
<h5>61.10-4 NaturalHomomorphismBySubspace</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NaturalHomomorphismBySubspace</code>( <var class="Arg">V</var>, <var class="Arg">W</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>For an <span class="SimpleMath">R</span>-vector space <var class="Arg">V</var> and a subspace <var class="Arg">W</var> of <var class="Arg">V</var>, <code class="func">NaturalHomomorphismBySubspace</code> returns the <span class="SimpleMath">R</span>-linear mapping that is the natural projection of <var class="Arg">V</var> onto the factor space <code class="code"><var class="Arg">V</var> / <var class="Arg">W</var></code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= Rationals^3;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">W:= VectorSpace( Rationals, [ [ 1, 1, 1 ] ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">f:= NaturalHomomorphismBySubspace( V, W );</span>
<linear mapping by matrix, ( Rationals^3 ) -> ( Rationals^2 )>
</pre></div>
<p><a id="X80015C78876B4F1E" name="X80015C78876B4F1E"></a></p>
<h5>61.10-5 Hom</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Hom</code>( <var class="Arg">F</var>, <var class="Arg">V</var>, <var class="Arg">W</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>For a field <var class="Arg">F</var> and two vector spaces <var class="Arg">V</var> and <var class="Arg">W</var> that can be regarded as <var class="Arg">F</var>-modules (see <code class="func">AsLeftModule</code> (<a href="chap57.html#X7EB3E46D7BC4A35C"><span class="RefLink">57.1-5</span></a>)), <code class="func">Hom</code> returns the <var class="Arg">F</var>-vector space of all <var class="Arg">F</var>-linear mappings from <var class="Arg">V</var> to <var class="Arg">W</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= Rationals^2;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">W:= VectorSpace( Rationals, [ [ 1, 0, 1 ], [ 1, 2, 3 ] ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">H:= Hom( Rationals, V, W );</span>
Hom( Rationals, ( Rationals^2 ), <vector space over Rationals, with
2 generators> )
<span class="GAPprompt">gap></span> <span class="GAPinput">Dimension( H );</span>
4
</pre></div>
<p><a id="X8680ADD381ECF879" name="X8680ADD381ECF879"></a></p>
<h5>61.10-6 End</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ End</code>( <var class="Arg">F</var>, <var class="Arg">V</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>For a field <var class="Arg">F</var> and a vector space <var class="Arg">V</var> that can be regarded as an <var class="Arg">F</var>-module (see <code class="func">AsLeftModule</code> (<a href="chap57.html#X7EB3E46D7BC4A35C"><span class="RefLink">57.1-5</span></a>)), <code class="func">End</code> returns the <var class="Arg">F</var>-algebra of all <var class="Arg">F</var>-linear mappings from <var class="Arg">V</var> to <var class="Arg">V</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= End( Rationals, Rationals^2 );</span>
End( Rationals, ( Rationals^2 ) )
<span class="GAPprompt">gap></span> <span class="GAPinput">Dimension( A );</span>
4
</pre></div>
<p><a id="X7A9A08EA79259659" name="X7A9A08EA79259659"></a></p>
<h5>61.10-7 IsFullHomModule</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsFullHomModule</code>( <var class="Arg">M</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>A <em>full hom module</em> is a module of all <span class="SimpleMath">R</span>-linear mappings between two left <span class="SimpleMath">R</span>-modules. The function <code class="func">Hom</code> (<a href="chap61.html#X80015C78876B4F1E"><span class="RefLink">61.10-5</span></a>) can be used to construct a full hom module.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= Rationals^2;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">W:= VectorSpace( Rationals, [ [ 1, 0, 1 ], [ 1, 2, 3 ] ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">H:= Hom( Rationals, V, W );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsFullHomModule( H );</span>
true
</pre></div>
<p><a id="X7C4737687E76A24A" name="X7C4737687E76A24A"></a></p>
<h5>61.10-8 IsPseudoCanonicalBasisFullHomModule</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPseudoCanonicalBasisFullHomModule</code>( <var class="Arg">B</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>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>
<p>Note that this is not a canonical basis (see <code class="func">CanonicalBasis</code> (<a href="chap61.html#X7C8EBFF5805F8C51"><span class="RefLink">61.5-3</span></a>)) because it depends on the stored bases of source and range.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPseudoCanonicalBasisFullHomModule( Basis( H ) );</span>
true
</pre></div>
<p><a id="X84F87C327A1856F2" name="X84F87C327A1856F2"></a></p>
<h5>61.10-9 IsLinearMappingsModule</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsLinearMappingsModule</code>( <var class="Arg">V</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<p>If an <span class="SimpleMath">F</span>-vector space <var class="Arg">V</var> is in the filter <code class="func">IsLinearMappingsModule</code> then this expresses that <var class="Arg">V</var> consists of linear mappings, and that <var class="Arg">V</var> is handled via the mechanism of nice bases (see <a href="chap61.html#X81503EB77FCE648D"><span class="RefLink">61.11</span></a>), in the following way. Let <span class="SimpleMath">S</span> and <span class="SimpleMath">R</span> be the source and the range, respectively, of each mapping in <span class="SimpleMath">V</span>. Then the <code class="func">NiceFreeLeftModuleInfo</code> (<a href="chap61.html#X79350786800C2DD8"><span class="RefLink">61.11-3</span></a>) value of <var class="Arg">V</var> is a record with the components <code class="code">basissource</code> (a basis <span class="SimpleMath">B_S</span> of <span class="SimpleMath">S</span>) and <code class="code">basisrange</code> (a basis <span class="SimpleMath">B_R</span> of <span class="SimpleMath">R</span>), and the <code class="func">NiceVector</code> (<a href="chap61.html#X807B8032780C59A4"><span class="RefLink">61.11-2</span></a>) value of <span class="SimpleMath">v ∈ <var class="Arg">V</var></span> is defined as the matrix of the <span class="SimpleMath">F</span>-linear mapping <span class="SimpleMath">v</span> w.r.t. the bases <span class="SimpleMath">B_S</span> and <span class="SimpleMath">B_R</span>.</p>
<p><a id="X81503EB77FCE648D" name="X81503EB77FCE648D"></a></p>
<h4>61.11 <span class="Heading">Vector Spaces Handled By Nice Bases</span></h4>
<p>There are kinds of free <span class="SimpleMath">R</span>-modules for which efficient computations are possible because the elements are <q>nice</q>, for example subspaces of full row modules or of full matrix modules. In other cases, a <q>nice</q> canonical basis is known that allows one to do the necessary computations in the corresponding row module, for example algebras given by structure constants.</p>
<p>In many other situations, one knows at least an isomorphism from the given module <span class="SimpleMath">V</span> to a <q>nicer</q> free left module <span class="SimpleMath">W</span>, in the sense that for each vector in <span class="SimpleMath">V</span>, the image in <span class="SimpleMath">W</span> can easily be computed, and analogously for each vector in <span class="SimpleMath">W</span>, one can compute the preimage in <span class="SimpleMath">V</span>.</p>
<p>This allows one to delegate computations w.r.t. a basis <span class="SimpleMath">B</span> of <span class="SimpleMath">V</span> to the corresponding basis <span class="SimpleMath">C</span> of <span class="SimpleMath">W</span>. We call <span class="SimpleMath">W</span> the <em>nice free left module</em> of <span class="SimpleMath">V</span>, and <span class="SimpleMath">C</span> the <em>nice basis</em> of <span class="SimpleMath">B</span>. (Note that it may happen that also <span class="SimpleMath">C</span> delegates questions to a <q>nicer</q> basis.) The basis <span class="SimpleMath">B</span> indicates the intended behaviour by the filter <code class="func">IsBasisByNiceBasis</code> (<a href="chap61.html#X82BC30A487967F5B"><span class="RefLink">61.11-5</span></a>), and stores <span class="SimpleMath">C</span> as value of the attribute <code class="func">NiceBasis</code> (<a href="chap61.html#X8388E0248690C214"><span class="RefLink">61.11-4</span></a>). <span class="SimpleMath">V</span> indicates the intended behaviour by the filter <code class="func">IsHandledByNiceBasis</code> (<a href="chap61.html#X79D1DEA679AEDA40"><span class="RefLink">61.11-6</span></a>), and stores <span class="SimpleMath">W</span> as value of the attribute <code class="func">NiceFreeLeftModule</code> (<a href="chap61.html#X826FD4BC7BA0559D"><span class="RefLink">61.11-1</span></a>).</p>
<p>The bijection between <span class="SimpleMath">V</span> and <span class="SimpleMath">W</span> is implemented by the functions <code class="func">NiceVector</code> (<a href="chap61.html#X807B8032780C59A4"><span class="RefLink">61.11-2</span></a>) and <code class="func">UglyVector</code> (<a href="chap61.html#X807B8032780C59A4"><span class="RefLink">61.11-2</span></a>); additional data needed to compute images and preimages can be stored as value of <code class="func">NiceFreeLeftModuleInfo</code> (<a href="chap61.html#X79350786800C2DD8"><span class="RefLink">61.11-3</span></a>).</p>
<p><a id="X826FD4BC7BA0559D" name="X826FD4BC7BA0559D"></a></p>
<h5>61.11-1 NiceFreeLeftModule</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NiceFreeLeftModule</code>( <var class="Arg">V</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>For a free left module <var class="Arg">V</var> that is handled via the mechanism of nice bases, this attribute stores the associated free left module to which the tasks are delegated.</p>
<p><a id="X807B8032780C59A4" name="X807B8032780C59A4"></a></p>
<h5>61.11-2 NiceVector</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NiceVector</code>( <var class="Arg">V</var>, <var class="Arg">v</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UglyVector</code>( <var class="Arg">V</var>, <var class="Arg">r</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><code class="func">NiceVector</code> and <code class="func">UglyVector</code> provide the linear bijection between the free left module <var class="Arg">V</var> and <code class="code"><var class="Arg">W</var>:= NiceFreeLeftModule( <var class="Arg">V</var> )</code>.</p>
<p>If <var class="Arg">v</var> lies in the elements family of the family of <var class="Arg">V</var> then <code class="code">NiceVector( <var class="Arg">v</var> )</code> is either <code class="keyw">fail</code> or an element in the elements family of the family of <var class="Arg">W</var>.</p>
<p>If <var class="Arg">r</var> lies in the elements family of the family of <var class="Arg">W</var> then <code class="code">UglyVector( <var class="Arg">r</var> )</code> is either <code class="keyw">fail</code> or an element in the elements family of the family of <var class="Arg">V</var>.</p>
<p>If <var class="Arg">v</var> lies in <var class="Arg">V</var> (which usually <em>cannot</em> be checked without using <var class="Arg">W</var>) then <code class="code">UglyVector( <var class="Arg">V</var>, NiceVector( <var class="Arg">V</var>, <var class="Arg">v</var> ) ) = <var class="Arg">v</var></code>. If <var class="Arg">r</var> lies in <var class="Arg">W</var> (which usually <em>can</em> be checked) then <code class="code">NiceVector( <var class="Arg">V</var>, UglyVector( <var class="Arg">V</var>, <var class="Arg">r</var> ) ) = <var class="Arg">r</var></code>.</p>
<p>(This allows one to implement for example a membership test for <var class="Arg">V</var> using the membership test in <var class="Arg">W</var>.)</p>
<p><a id="X79350786800C2DD8" name="X79350786800C2DD8"></a></p>
<h5>61.11-3 NiceFreeLeftModuleInfo</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NiceFreeLeftModuleInfo</code>( <var class="Arg">V</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>For a free left module <var class="Arg">V</var> that is handled via the mechanism of nice bases, this operation has to provide the necessary information (if any) for calls of <code class="func">NiceVector</code> (<a href="chap61.html#X807B8032780C59A4"><span class="RefLink">61.11-2</span></a>) and <code class="func">UglyVector</code> (<a href="chap61.html#X807B8032780C59A4"><span class="RefLink">61.11-2</span></a>).</p>
<p><a id="X8388E0248690C214" name="X8388E0248690C214"></a></p>
<h5>61.11-4 NiceBasis</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NiceBasis</code>( <var class="Arg">B</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Let <var class="Arg">B</var> be a basis of a free left module <var class="Arg">V</var> that is handled via nice bases. If <var class="Arg">B</var> has no basis vectors stored at the time of the first call to <code class="code">NiceBasis</code> then <code class="code">NiceBasis( <var class="Arg">B</var> )</code> is obtained as <code class="code">Basis( NiceFreeLeftModule( <var class="Arg">V</var> ) )</code>. If basis vectors are stored then <code class="code">NiceBasis( <var class="Arg">B</var> )</code> is the result of the call of <code class="code">Basis</code> with arguments <code class="code">NiceFreeLeftModule( <var class="Arg">V</var> )</code> and the <code class="code">NiceVector</code> values of the basis vectors of <var class="Arg">B</var>.</p>
<p>Note that the result is <code class="keyw">fail</code> if and only if the <q>basis vectors</q> stored in <var class="Arg">B</var> are in fact not basis vectors.</p>
<p>The attributes <code class="code">GeneratorsOfLeftModule</code> of the underlying left modules of <var class="Arg">B</var> and the result of <code class="code">NiceBasis</code> correspond via <code class="func">NiceVector</code> (<a href="chap61.html#X807B8032780C59A4"><span class="RefLink">61.11-2</span></a>) and <code class="func">UglyVector</code> (<a href="chap61.html#X807B8032780C59A4"><span class="RefLink">61.11-2</span></a>).</p>
<p><a id="X82BC30A487967F5B" name="X82BC30A487967F5B"></a></p>
<h5>61.11-5 IsBasisByNiceBasis</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsBasisByNiceBasis</code>( <var class="Arg">B</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>This filter indicates that the basis <var class="Arg">B</var> delegates tasks such as the computation of coefficients (see <code class="func">Coefficients</code> (<a href="chap61.html#X80B32F667BF6AFD8"><span class="RefLink">61.6-3</span></a>)) to a basis of an isomorphic <q>nicer</q> free left module.</p>
<p><a id="X79D1DEA679AEDA40" name="X79D1DEA679AEDA40"></a></p>
<h5>61.11-6 IsHandledByNiceBasis</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsHandledByNiceBasis</code>( <var class="Arg">M</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>For a free left module <var class="Arg">M</var> in this category, essentially all operations are performed using a <q>nicer</q> free left module, which is usually a row module.</p>
<p><a id="X8238195B851D3C44" name="X8238195B851D3C44"></a></p>
<h4>61.12 <span class="Heading">How to Implement New Kinds of Vector Spaces</span></h4>
<p><a id="X7DE34C3E837FCBC3" name="X7DE34C3E837FCBC3"></a></p>
<h5>61.12-1 DeclareHandlingByNiceBasis</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DeclareHandlingByNiceBasis</code>( <var class="Arg">name</var>, <var class="Arg">info</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InstallHandlingByNiceBasis</code>( <var class="Arg">name</var>, <var class="Arg">record</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>These functions are used to implement a new kind of free left modules that shall be handled via the mechanism of nice bases (see <a href="chap61.html#X81503EB77FCE648D"><span class="RefLink">61.11</span></a>).</p>
<p><var class="Arg">name</var> must be a string, a filter <span class="SimpleMath">f</span> with this name is created which implies <code class="func">IsFreeLeftModule</code> (<a href="chap57.html#X7C4832187F3D9228"><span class="RefLink">57.3-1</span></a>), and a logical implication from the join of <span class="SimpleMath">f</span> with <code class="func">IsAttributeStoringRep</code> (<a href="chap13.html#X7A951C33839AF2C1"><span class="RefLink">13.5-5</span></a>) to <code class="func">IsHandledByNiceBasis</code> (<a href="chap61.html#X79D1DEA679AEDA40"><span class="RefLink">61.11-6</span></a>) is installed.</p>
<p><var class="Arg">record</var> must be a record with the following components.</p>
<dl>
<dt><strong class="Mark"><code class="code">detect</code> </strong></dt>
<dd><p>a function of four arguments <span class="SimpleMath">R</span>, <span class="SimpleMath">l</span>, <span class="SimpleMath">V</span>, and <span class="SimpleMath">z</span>, where <span class="SimpleMath">V</span> is a free left module over the ring <span class="SimpleMath">R</span> with generators the list or collection <span class="SimpleMath">l</span>, and <span class="SimpleMath">z</span> is either the zero element of <span class="SimpleMath">V</span> or <code class="keyw">false</code> (then <span class="SimpleMath">l</span> is nonempty); the function returns <code class="keyw">true</code> if <span class="SimpleMath">V</span> shall lie in the filter <span class="SimpleMath">f</span>, and <code class="keyw">false</code> otherwise; the return value may also be <code class="keyw">fail</code>, which indicates that <span class="SimpleMath">V</span> is <em>not</em> to be handled via the mechanism of nice bases at all,</p>
</dd>
<dt><strong class="Mark"><code class="code">NiceFreeLeftModuleInfo</code> </strong></dt>
<dd><p>the <code class="func">NiceFreeLeftModuleInfo</code> (<a href="chap61.html#X79350786800C2DD8"><span class="RefLink">61.11-3</span></a>) method for left modules in <span class="SimpleMath">f</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">NiceVector</code> </strong></dt>
<dd><p>the <code class="func">NiceVector</code> (<a href="chap61.html#X807B8032780C59A4"><span class="RefLink">61.11-2</span></a>) method for left modules <span class="SimpleMath">V</span> in <span class="SimpleMath">f</span>; called with <span class="SimpleMath">V</span> and a vector <span class="SimpleMath">v ∈ V</span>, this function returns the nice vector <span class="SimpleMath">r</span> associated with <span class="SimpleMath">v</span>, and</p>
</dd>
<dt><strong class="Mark"><code class="code">UglyVector</code></strong></dt>
<dd><p>the <code class="func">UglyVector</code> (<a href="chap61.html#X807B8032780C59A4"><span class="RefLink">61.11-2</span></a>) method for left modules <span class="SimpleMath">V</span> in <span class="SimpleMath">f</span>; called with <span class="SimpleMath">V</span> and a vector <span class="SimpleMath">r</span> in the <code class="func">NiceFreeLeftModule</code> (<a href="chap61.html#X826FD4BC7BA0559D"><span class="RefLink">61.11-1</span></a>) value of <span class="SimpleMath">V</span>, this function returns the vector <span class="SimpleMath">v ∈ V</span> to which <span class="SimpleMath">r</span> is associated.</p>
</dd>
</dl>
<p>The idea is that all one has to do for implementing a new kind of free left modules handled by the mechanism of nice bases is to call <code class="func">DeclareHandlingByNiceBasis</code> and <code class="func">InstallHandlingByNiceBasis</code>, which causes the installation of the necessary methods and adds the pair <span class="SimpleMath">[ f,</span><var class="Arg">record</var><code class="code">.detect</code><span class="SimpleMath">]</span> to the global list <code class="func">NiceBasisFiltersInfo</code> (<a href="chap61.html#X7E6077F0830A28DA"><span class="RefLink">61.12-2</span></a>). The <code class="func">LeftModuleByGenerators</code> (<a href="chap57.html#X79ED1D7D7F0AE59A"><span class="RefLink">57.1-10</span></a>) methods call <code class="func">CheckForHandlingByNiceBasis</code> (<a href="chap61.html#X7A374553786DF5E7"><span class="RefLink">61.12-3</span></a>), which sets the appropriate filter for the desired left module if applicable.</p>
<p><a id="X7E6077F0830A28DA" name="X7E6077F0830A28DA"></a></p>
<h5>61.12-2 NiceBasisFiltersInfo</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NiceBasisFiltersInfo</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p>An overview of all kinds of vector spaces that are currently handled by nice bases is given by the global list <code class="code">NiceBasisFiltersInfo</code>. Examples of such vector spaces are vector spaces of field elements (but not the fields themselves) and non-Gaussian row and matrix spaces (see <code class="func">IsGaussianSpace</code> (<a href="chap61.html#X83724C157F4FDFB4"><span class="RefLink">61.9-3</span></a>)).</p>
<p><a id="X7A374553786DF5E7" name="X7A374553786DF5E7"></a></p>
<h5>61.12-3 CheckForHandlingByNiceBasis</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CheckForHandlingByNiceBasis</code>( <var class="Arg">R</var>, <var class="Arg">gens</var>, <var class="Arg">M</var>, <var class="Arg">zero</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Whenever a free left module is constructed for which the filter <code class="code">IsHandledByNiceBasis</code> may be useful, <code class="code">CheckForHandlingByNiceBasis</code> should be called. (This is done in the methods for <code class="code">VectorSpaceByGenerators</code>, <code class="code">AlgebraByGenerators</code>, <code class="code">IdealByGenerators</code> etc. in the <strong class="pkg">GAP</strong> library.)</p>
<p>The arguments of this function are the coefficient ring <var class="Arg">R</var>, the list <var class="Arg">gens</var> of generators, the constructed module <var class="Arg">M</var> itself, and the zero element <var class="Arg">zero</var> of <var class="Arg">M</var>; if <var class="Arg">gens</var> is nonempty then the <var class="Arg">zero</var> value may also be <code class="keyw">false</code>.</p>
<p><a id="X78515F448644204E" name="X78515F448644204E"></a></p>
<h4>61.13 <span class="Heading">Tensor Products and Exterior and Symmetric Powers</span></h4>
<p><a id="X81B2276A7EBA8ED1" name="X81B2276A7EBA8ED1"></a></p>
<h5>61.13-1 TensorProduct</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TensorProduct</code>( <var class="Arg">list</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TensorProduct</code>( <var class="Arg">V</var>, <var class="Arg">W</var>, <var class="Arg">...</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Here <var class="Arg">list</var> must be a list of vector spaces. This function returns the tensor product of the elements in the list. The vector spaces must be defined over the same field.</p>
<p>In the second form, the vector spaces are given individually.</p>
<p>Elements of the tensor product <span class="SimpleMath">V_1⊗ ⋯ ⊗ V_k</span> are linear combinations of <span class="SimpleMath">v_1⊗⋯ ⊗ v_k</span>, where the <span class="SimpleMath">v_i</span> are arbitrary basis elements of <span class="SimpleMath">V_i</span>. In <strong class="pkg">GAP</strong> a tensor element like that is printed as</p>
<div class="example"><pre>
v_1<x> ... <x>v_k
</pre></div>
<p>Furthermore, the zero of a tensor product is printed as</p>
<div class="example"><pre>
<0-tensor>
</pre></div>
<p>This does not mean that all tensor products have the same zero element: zeros of different tensor products have different families.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:=TensorProduct(Rationals^2, Rationals^3);</span>
<vector space over Rationals, with 6 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">Basis(V);</span>
Basis( <vector space over Rationals, with 6 generators>,
[ 1*([ 0, 1 ]<x>[ 0, 0, 1 ]), 1*([ 0, 1 ]<x>[ 0, 1, 0 ]),
1*([ 0, 1 ]<x>[ 1, 0, 0 ]), 1*([ 1, 0 ]<x>[ 0, 0, 1 ]),
1*([ 1, 0 ]<x>[ 0, 1, 0 ]), 1*([ 1, 0 ]<x>[ 1, 0, 0 ]) ] )
</pre></div>
<p>See also <code class="func">KroneckerProduct</code> (<a href="chap24.html#X8634C79E7DB22934"><span class="RefLink">24.5-9</span></a>).</p>
<p><a id="X787BB7FF85F0AD68" name="X787BB7FF85F0AD68"></a></p>
<h5>61.13-2 ExteriorPower</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ExteriorPower</code>( <var class="Arg">V</var>, <var class="Arg">k</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Here <var class="Arg">V</var> must be a vector space. This function returns the <var class="Arg">k</var>-th exterior power of <var class="Arg">V</var>.</p>
<p>Elements of the exterior power <span class="SimpleMath">⋀^k V</span> are linear combinations of <span class="SimpleMath">v_i_1∧⋯ ∧ v_i_k</span>, where the <span class="SimpleMath">v_i_j</span> are basis elements of <span class="SimpleMath">V</span>, and <span class="SimpleMath">1 ≤ i_1 < i_2 ⋯ < i_k</span>. In <strong class="pkg">GAP</strong> a wedge element like that is printed as</p>
<div class="example"><pre>
v_1/\ ... /\v_k
</pre></div>
<p>Furthermore, the zero of an exterior power is printed as</p>
<div class="example"><pre>
<0-wedge>
</pre></div>
<p>This does not mean that all exterior powers have the same zero element: zeros of different exterior powers have different families.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:=ExteriorPower(Rationals^3, 2);</span>
<vector space of dimension 3 over Rationals>
<span class="GAPprompt">gap></span> <span class="GAPinput">Basis(V);</span>
Basis( <vector space of dimension 3 over Rationals>, [
1*([ 0, 1, 0 ]/\[ 0, 0, 1 ]), 1*([ 1, 0, 0 ]/\[ 0, 0, 1 ]),
1*([ 1, 0, 0 ]/\[ 0, 1, 0 ]) ] )
</pre></div>
<p><a id="X79E2C2AF842E8419" name="X79E2C2AF842E8419"></a></p>
<h5>61.13-3 SymmetricPower</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SymmetricPower</code>( <var class="Arg">V</var>, <var class="Arg">k</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Here <var class="Arg">V</var> must be a vector space. This function returns the <var class="Arg">k</var>-th symmetric power of <var class="Arg">V</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:=SymmetricPower(Rationals^3, 2);</span>
<vector space over Rationals, with 6 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">Basis(V);</span>
Basis( <vector space over Rationals, with 6 generators>,
[ 1*([ 0, 0, 1 ].[ 0, 0, 1 ]), 1*([ 0, 1, 0 ].[ 0, 0, 1 ]),
1*([ 0, 1, 0 ].[ 0, 1, 0 ]), 1*([ 1, 0, 0 ].[ 0, 0, 1 ]),
1*([ 1, 0, 0 ].[ 0, 1, 0 ]), 1*([ 1, 0, 0 ].[ 1, 0, 0 ])
] )
</pre></div>
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