  
  [1X61 [33X[0;0YVector Spaces[133X[101X
  
  
  [1X61.1 [33X[0;0YIsLeftVectorSpace (Filter)[133X[101X
  
  [1X61.1-1 IsLeftVectorSpace[101X
  
  [33X[1;0Y[29X[2XIsLeftVectorSpace[102X( [3XV[103X ) [32X Category[133X
  [33X[1;0Y[29X[2XIsVectorSpace[102X( [3XV[103X ) [32X Category[133X
  
  [33X[0;0YA  [13Xvector space[113X in [5XGAP[105X is a free left module (see [2XIsFreeLeftModule[102X ([14X57.3-1[114X))
  over a division ring (see Chapter [14X58[114X).[133X
  
  [33X[0;0YWhenever  we  talk  about  an  [22XF[122X-vector  space [3XV[103X then [3XV[103X is an additive group
  (see [2XIsAdditiveGroup[102X  ([14X55.1-6[114X))  on  which  the  division  ring  [22XF[122X  acts via
  multiplication from the left such that this action and the addition in [3XV[103X are
  left and right distributive. The division ring [22XF[122X can be accessed as value of
  the attribute [2XLeftActingDomain[102X ([14X57.1-11[114X).[133X
  
  [33X[0;0YVector  spaces  in  [5XGAP[105X are always [13Xleft[113X vector spaces, [2XIsLeftVectorSpace[102X and
  [2XIsVectorSpace[102X are synonyms.[133X
  
  
  [1X61.2 [33X[0;0YConstructing Vector Spaces[133X[101X
  
  [1X61.2-1 VectorSpace[101X
  
  [33X[1;0Y[29X[2XVectorSpace[102X( [3XF[103X, [3Xgens[103X[, [3Xzero[103X][, [3X"basis"[103X] ) [32X function[133X
  
  [33X[0;0YFor  a  field  [3XF[103X  and  a collection [3Xgens[103X of vectors, [2XVectorSpace[102X returns the
  [3XF[103X-vector space spanned by the elements in [3Xgens[103X.[133X
  
  [33X[0;0YThe  optional  argument  [3Xzero[103X can be used to specify the zero element of the
  space;  [3Xzero[103X  [13Xmust[113X  be  given  if [3Xgens[103X is empty. The optional string [10X"basis"[110X
  indicates  that  [3Xgens[103X  is  known  to  be  linearly  independent  over  [3XF[103X, in
  particular  the  dimension of the vector space is immediately set; note that
  [2XBasis[102X  ([14X61.5-2[114X)  need  [13Xnot[113X  return  the  basis  formed by [3Xgens[103X if the string
  [10X"basis"[110X is given as an argument.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XV:= VectorSpace( Rationals, [ [ 1, 2, 3 ], [ 1, 1, 1 ] ] );[127X[104X
    [4X[28X<vector space over Rationals, with 2 generators>[128X[104X
  [4X[32X[104X
  
  [1X61.2-2 Subspace[101X
  
  [33X[1;0Y[29X[2XSubspace[102X( [3XV[103X, [3Xgens[103X[, [3X"basis"[103X] ) [32X function[133X
  [33X[1;0Y[29X[2XSubspaceNC[102X( [3XV[103X, [3Xgens[103X[, [3X"basis"[103X] ) [32X function[133X
  
  [33X[0;0YFor an [22XF[122X-vector space [3XV[103X and a list or collection [3Xgens[103X that is a subset of [3XV[103X,
  [2XSubspace[102X  returns  the [22XF[122X-vector space spanned by [3Xgens[103X; if [3Xgens[103X is empty then
  the  trivial  subspace  (see [2XTrivialSubspace[102X ([14X61.3-2[114X)) of [3XV[103X is returned. The
  parent (see [14X31.7[114X) of the returned vector space is set to [3XV[103X.[133X
  
  [33X[0;0Y[2XSubspaceNC[102X does the same as [2XSubspace[102X, except that it omits the check whether
  [3Xgens[103X is a subset of [3XV[103X.[133X
  
  [33X[0;0YThe  optional  string  [3X"basis"[103X  indicates  that [3Xgens[103X is known to be linearly
  independent  over  [22XF[122X.  In  this  case  the  dimension  of  the  subspace  is
  immediately  set, and both [2XSubspace[102X and [2XSubspaceNC[102X do [13Xnot[113X check whether [3Xgens[103X
  really is linearly independent and whether [3Xgens[103X is a subset of [3XV[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XV:= VectorSpace( Rationals, [ [ 1, 2, 3 ], [ 1, 1, 1 ] ] );;[127X[104X
    [4X[25Xgap>[125X [27XW:= Subspace( V, [ [ 0, 1, 2 ] ] );[127X[104X
    [4X[28X<vector space over Rationals, with 1 generator>[128X[104X
  [4X[32X[104X
  
  [1X61.2-3 AsVectorSpace[101X
  
  [33X[1;0Y[29X[2XAsVectorSpace[102X( [3XF[103X, [3XD[103X ) [32X operation[133X
  
  [33X[0;0YLet  [3XF[103X  be  a  division  ring  and  [3XD[103X a domain. If the elements in [3XD[103X form an
  [3XF[103X-vector  space  then  [2XAsVectorSpace[102X  returns this [3XF[103X-vector space, otherwise
  [9Xfail[109X is returned.[133X
  
  [33X[0;0Y[2XAsVectorSpace[102X  can  be  used  for  example to view a given vector space as a
  vector space over a smaller or larger division ring.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XV:= FullRowSpace( GF( 27 ), 3 );[127X[104X
    [4X[28X( GF(3^3)^3 )[128X[104X
    [4X[25Xgap>[125X [27XDimension( V );  LeftActingDomain( V );[127X[104X
    [4X[28X3[128X[104X
    [4X[28XGF(3^3)[128X[104X
    [4X[25Xgap>[125X [27XW:= AsVectorSpace( GF( 3 ), V );[127X[104X
    [4X[28X<vector space over GF(3), with 9 generators>[128X[104X
    [4X[25Xgap>[125X [27XDimension( W );  LeftActingDomain( W );[127X[104X
    [4X[28X9[128X[104X
    [4X[28XGF(3)[128X[104X
    [4X[25Xgap>[125X [27XAsVectorSpace( GF( 9 ), V );[127X[104X
    [4X[28Xfail[128X[104X
  [4X[32X[104X
  
  [1X61.2-4 AsSubspace[101X
  
  [33X[1;0Y[29X[2XAsSubspace[102X( [3XV[103X, [3XU[103X ) [32X operation[133X
  
  [33X[0;0YLet  [3XV[103X  be an [22XF[122X-vector space, and [3XU[103X a collection. If [3XU[103X is a subset of [3XV[103X such
  that  the  elements of [3XU[103X form an [22XF[122X-vector space then [2XAsSubspace[102X returns this
  vector  space,  with parent set to [3XV[103X (see [2XAsVectorSpace[102X ([14X61.2-3[114X)). Otherwise
  [9Xfail[109X is returned.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XV:= VectorSpace( Rationals, [ [ 1, 2, 3 ], [ 1, 1, 1 ] ] );;[127X[104X
    [4X[25Xgap>[125X [27XW:= VectorSpace( Rationals, [ [ 1/2, 1/2, 1/2 ] ] );;[127X[104X
    [4X[25Xgap>[125X [27XU:= AsSubspace( V, W );[127X[104X
    [4X[28X<vector space over Rationals, with 1 generator>[128X[104X
    [4X[25Xgap>[125X [27XParent( U ) = V;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XAsSubspace( V, [ [ 1, 1, 1 ] ] );[127X[104X
    [4X[28Xfail[128X[104X
  [4X[32X[104X
  
  
  [1X61.3 [33X[0;0YOperations and Attributes for Vector Spaces[133X[101X
  
  [1X61.3-1 GeneratorsOfLeftVectorSpace[101X
  
  [33X[1;0Y[29X[2XGeneratorsOfLeftVectorSpace[102X( [3XV[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XGeneratorsOfVectorSpace[102X( [3XV[103X ) [32X attribute[133X
  
  [33X[0;0YFor  an  [22XF[122X-vector  space  [3XV[103X,  [2XGeneratorsOfLeftVectorSpace[102X  returns a list of
  vectors in [3XV[103X that generate [3XV[103X as an [22XF[122X-vector space.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XGeneratorsOfVectorSpace( FullRowSpace( Rationals, 3 ) );[127X[104X
    [4X[28X[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ][128X[104X
  [4X[32X[104X
  
  [1X61.3-2 TrivialSubspace[101X
  
  [33X[1;0Y[29X[2XTrivialSubspace[102X( [3XV[103X ) [32X attribute[133X
  
  [33X[0;0YFor  a  vector  space  [3XV[103X,  [2XTrivialSubspace[102X  returns  the  subspace of [3XV[103X that
  consists of the zero vector in [3XV[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XV:= GF(3)^3;;[127X[104X
    [4X[25Xgap>[125X [27Xtriv:= TrivialSubspace( V );[127X[104X
    [4X[28X<vector space of dimension 0 over GF(3)>[128X[104X
    [4X[25Xgap>[125X [27XAsSet( triv );[127X[104X
    [4X[28X[ [ 0*Z(3), 0*Z(3), 0*Z(3) ] ][128X[104X
  [4X[32X[104X
  
  
  [1X61.4 [33X[0;0YDomains of Subspaces of Vector Spaces[133X[101X
  
  [1X61.4-1 Subspaces[101X
  
  [33X[1;0Y[29X[2XSubspaces[102X( [3XV[103X[, [3Xk[103X] ) [32X attribute[133X
  
  [33X[0;0YCalled  with  a  finite  vector space [3Xv[103X, [2XSubspaces[102X returns the domain of all
  subspaces of [3XV[103X.[133X
  
  [33X[0;0YCalled  with  [3XV[103X and a nonnegative integer [3Xk[103X, [2XSubspaces[102X returns the domain of
  all [3Xk[103X-dimensional subspaces of [3XV[103X.[133X
  
  [33X[0;0YSpecial  [2XSize[102X  ([14X30.4-6[114X) and [2XIterator[102X ([14X30.8-1[114X) methods are provided for these
  domains.[133X
  
  [1X61.4-2 IsSubspacesVectorSpace[101X
  
  [33X[1;0Y[29X[2XIsSubspacesVectorSpace[102X( [3XD[103X ) [32X Category[133X
  
  [33X[0;0YThe  domain  of all subspaces of a (finite) vector space or of all subspaces
  of  fixed  dimension,  as  returned  by  [2XSubspaces[102X  ([14X61.4-1[114X)  (see [2XSubspaces[102X
  ([14X61.4-1[114X)) lies in the category [2XIsSubspacesVectorSpace[102X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD:= Subspaces( GF(3)^3 );[127X[104X
    [4X[28XSubspaces( ( GF(3)^3 ) )[128X[104X
    [4X[25Xgap>[125X [27XSize( D );[127X[104X
    [4X[28X28[128X[104X
    [4X[25Xgap>[125X [27Xiter:= Iterator( D );;[127X[104X
    [4X[25Xgap>[125X [27XNextIterator( iter );[127X[104X
    [4X[28X<vector space of dimension 0 over GF(3)>[128X[104X
    [4X[25Xgap>[125X [27XNextIterator( iter );[127X[104X
    [4X[28X<vector space of dimension 1 over GF(3)>[128X[104X
    [4X[25Xgap>[125X [27XIsSubspacesVectorSpace( D );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X61.5 [33X[0;0YBases of Vector Spaces[133X[101X
  
  [33X[0;0YIn  [5XGAP[105X,  a  [13Xbasis[113X of a free left [22XF[122X-module [22XV[122X is a list of vectors [22XB = [ v_1,
  v_2,  ...,  v_n  ][122X in [22XV[122X such that [22XV[122X is generated as a left [22XF[122X-module by these
  vectors and such that [22XB[122X is linearly independent over [22XF[122X. The integer [22Xn[122X is the
  dimension  of  [22XV[122X (see [2XDimension[102X ([14X57.3-3[114X)). In particular, as each basis is a
  list  (see Chapter [14X21[114X), it has a length (see [2XLength[102X ([14X21.17-5[114X)), and the [22Xi[122X-th
  vector of [22XB[122X can be accessed as [22XB[i][122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XV:= Rationals^3;[127X[104X
    [4X[28X( Rationals^3 )[128X[104X
    [4X[25Xgap>[125X [27XB:= Basis( V );[127X[104X
    [4X[28XCanonicalBasis( ( Rationals^3 ) )[128X[104X
    [4X[25Xgap>[125X [27XLength( B );[127X[104X
    [4X[28X3[128X[104X
    [4X[25Xgap>[125X [27XB[1];[127X[104X
    [4X[28X[ 1, 0, 0 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  operations  described  below  make  sense  only  for  bases  of  [13Xfinite[113X
  dimensional  vector  spaces.  (In practice this means that the vector spaces
  must  be  [13Xlow[113X  dimensional,  that  is, the dimension should not exceed a few
  hundred.)[133X
  
  [33X[0;0YBesides  the basic operations for lists (see [14X21.2[114X), the [13Xbasic operations for
  bases[113X  are  [2XBasisVectors[102X  ([14X61.6-1[114X), [2XCoefficients[102X ([14X61.6-3[114X), [2XLinearCombination[102X
  ([14X61.6-4[114X),  and [2XUnderlyingLeftModule[102X ([14X61.6-2[114X). These and other operations for
  arbitrary bases are described in [14X61.6[114X.[133X
  
  [33X[0;0YFor special kinds of bases, further operations are defined (see [14X61.7[114X).[133X
  
  [33X[0;0Y[5XGAP[105X supports the following three kinds of bases.[133X
  
  [33X[0;0Y[13XRelative  bases[113X  delegate  the  work  to  other  bases of the same free left
  module, via basechange matrices (see [2XRelativeBasis[102X ([14X61.5-4[114X)).[133X
  
  [33X[0;0Y[13XBases  handled  by  nice bases[113X delegate the work to bases of isomorphic left
  modules over the same left acting domain (see [14X61.11[114X).[133X
  
  [33X[0;0YFinally, of course there must be bases in [5XGAP[105X that really do the work.[133X
  
  [33X[0;0YFor  example,  in  the  case of a Gaussian row or matrix space [3XV[103X (see [14X61.9[114X),
  [10XBasis( [3XV[103X[10X )[110X is a semi-echelonized basis (see [2XIsSemiEchelonized[102X ([14X61.9-7[114X)) that
  uses  Gaussian  elimination;  such  a  basis is of the third kind. [10XBasis( [3XV[103X[10X,
  [3Xvectors[103X[10X  )[110X 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.[133X
  
  [33X[0;0YBases   handled   by   nice  bases  are  described  in [14X61.11[114X.  Examples  are
  non-Gaussian  row  and  matrix  spaces,  and  subspaces of finite fields and
  abelian number fields that are themselves not fields.[133X
  
  [1X61.5-1 IsBasis[101X
  
  [33X[1;0Y[29X[2XIsBasis[102X( [3Xobj[103X ) [32X Category[133X
  
  [33X[0;0YIn [5XGAP[105X, a [13Xbasis[113X of a free left module is an object that knows how to compute
  coefficients w.r.t. its basis vectors (see [2XCoefficients[102X ([14X61.6-3[114X)). Bases are
  constructed  by  [2XBasis[102X  ([14X61.5-2[114X).  Each basis is an immutable list, the [22Xi[122X-th
  entry being the [22Xi[122X-th basis vector.[133X
  
  [33X[0;0Y(See [14X61.8[114X for mutable bases.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XV:= GF(2)^2;;[127X[104X
    [4X[25Xgap>[125X [27XB:= Basis( V );;[127X[104X
    [4X[25Xgap>[125X [27XIsBasis( B );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsBasis( [ [ 1, 0 ], [ 0, 1 ] ] );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsBasis( Basis( Rationals^2, [ [ 1, 0 ], [ 0, 1 ] ] ) );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X61.5-2 Basis[101X
  
  [33X[1;0Y[29X[2XBasis[102X( [3XV[103X[, [3Xvectors[103X] ) [32X attribute[133X
  [33X[1;0Y[29X[2XBasisNC[102X( [3XV[103X, [3Xvectors[103X ) [32X operation[133X
  
  [33X[0;0YCalled  with  a  free left [22XF[122X-module [3XV[103X as the only argument, [2XBasis[102X returns an
  [22XF[122X-basis of [3XV[103X whose vectors are not further specified.[133X
  
  [33X[0;0YIf  additionally  a  list  [3Xvectors[103X  of  vectors  in [3XV[103X is given that forms an
  [22XF[122X-basis  of  [3XV[103X  then  [2XBasis[102X  returns  this basis; if [3Xvectors[103X is not linearly
  independent  over [22XF[122X or does not generate [3XV[103X as a free left [22XF[122X-module then [9Xfail[109X
  is returned.[133X
  
  [33X[0;0Y[2XBasisNC[102X  does  the same as the two argument version of [2XBasis[102X, except that it
  does not check whether [3Xvectors[103X form a basis.[133X
  
  [33X[0;0YIf  no  basis  vectors  are  prescribed  then  [2XBasis[102X  need not compute basis
  vectors;  in  this  case,  the  vectors  are  computed  in the first call to
  [2XBasisVectors[102X ([14X61.6-1[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XV:= VectorSpace( Rationals, [ [ 1, 2, 7 ], [ 1/2, 1/3, 5 ] ] );;[127X[104X
    [4X[25Xgap>[125X [27XB:= Basis( V );[127X[104X
    [4X[28XSemiEchelonBasis( <vector space over Rationals, with [128X[104X
    [4X[28X2 generators>, ... )[128X[104X
    [4X[25Xgap>[125X [27XBasisVectors( B );[127X[104X
    [4X[28X[ [ 1, 2, 7 ], [ 0, 1, -9/4 ] ][128X[104X
    [4X[25Xgap>[125X [27XB:= Basis( V, [ [ 1, 2, 7 ], [ 3, 2, 30 ] ] );[127X[104X
    [4X[28XBasis( <vector space over Rationals, with 2 generators>, [128X[104X
    [4X[28X[ [ 1, 2, 7 ], [ 3, 2, 30 ] ] )[128X[104X
    [4X[25Xgap>[125X [27XBasis( V, [ [ 1, 2, 3 ] ] );[127X[104X
    [4X[28Xfail[128X[104X
  [4X[32X[104X
  
  [1X61.5-3 CanonicalBasis[101X
  
  [33X[1;0Y[29X[2XCanonicalBasis[102X( [3XV[103X ) [32X attribute[133X
  
  [33X[0;0YIf the vector space [3XV[103X supports a [13Xcanonical basis[113X then [2XCanonicalBasis[102X returns
  this basis, otherwise [9Xfail[109X is returned.[133X
  
  [33X[0;0YThe  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 [2XLeftActingDomain[102X ([14X57.1-11[114X))
  then  the  equality  of  these vector spaces can be decided by comparing the
  canonical bases.[133X
  
  [33X[0;0YThe  exact  meaning of a canonical basis depends on the type of [3XV[103X. Canonical
  bases are defined for example for Gaussian row and matrix spaces (see [14X61.9[114X).[133X
  
  [33X[0;0YIf  one  designs  a  new  kind  of  vector  spaces (see [14X61.12[114X) and defines a
  canonical basis for these spaces then the [2XCanonicalBasis[102X method one installs
  (see [2XInstallMethod[102X  ([14X78.3-1[114X))  must  [13Xnot[113X  call  [2XBasis[102X ([14X61.5-2[114X). On the other
  hand,  one probably should install a [2XBasis[102X ([14X61.5-2[114X) method that simply calls
  [2XCanonicalBasis[102X,   the   value  of  the  method  (see [14X78.3[114X  and  [14X78.4[114X)  being
  [10XCANONICAL_BASIS_FLAGS[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xvecs:= [ [ 1, 2, 3 ], [ 1, 1, 1 ], [ 1, 1, 1 ] ];;[127X[104X
    [4X[25Xgap>[125X [27XV:= VectorSpace( Rationals, vecs );;[127X[104X
    [4X[25Xgap>[125X [27XB:= CanonicalBasis( V );[127X[104X
    [4X[28XCanonicalBasis( <vector space over Rationals, with 3 generators> )[128X[104X
    [4X[25Xgap>[125X [27XBasisVectors( B );[127X[104X
    [4X[28X[ [ 1, 0, -1 ], [ 0, 1, 2 ] ][128X[104X
  [4X[32X[104X
  
  [1X61.5-4 RelativeBasis[101X
  
  [33X[1;0Y[29X[2XRelativeBasis[102X( [3XB[103X, [3Xvectors[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XRelativeBasisNC[102X( [3XB[103X, [3Xvectors[103X ) [32X operation[133X
  
  [33X[0;0YA  relative  basis  is  a basis of the free left module [3XV[103X that delegates the
  computation  of  coefficients  etc.  to  another basis of [3XV[103X via a basechange
  matrix.[133X
  
  [33X[0;0YLet [3XB[103X be a basis of the free left module [3XV[103X, and [3Xvectors[103X a list of vectors in
  [3XV[103X.[133X
  
  [33X[0;0Y[2XRelativeBasis[102X  checks  whether [3Xvectors[103X form a basis of [3XV[103X, and in this case a
  basis  is returned in which [3Xvectors[103X are the basis vectors; otherwise [9Xfail[109X is
  returned.[133X
  
  [33X[0;0Y[2XRelativeBasisNC[102X does the same, except that it omits the check.[133X
  
  
  [1X61.6 [33X[0;0YOperations for Vector Space Bases[133X[101X
  
  [1X61.6-1 BasisVectors[101X
  
  [33X[1;0Y[29X[2XBasisVectors[102X( [3XB[103X ) [32X attribute[133X
  
  [33X[0;0YFor  a  vector space basis [3XB[103X, [10XBasisVectors[110X returns the list of basis vectors
  of  [3XB[103X.  The  lists  [3XB[103X  and  [10XBasisVectors( [3XB[103X[10X )[110X are equal; the main purpose of
  [10XBasisVectors[110X  is  to  provide access to a list of vectors that does [13Xnot[113X know
  about an underlying vector space.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XV:= VectorSpace( Rationals, [ [ 1, 2, 7 ], [ 1/2, 1/3, 5 ] ] );;[127X[104X
    [4X[25Xgap>[125X [27XB:= Basis( V, [ [ 1, 2, 7 ], [ 0, 1, -9/4 ] ] );;[127X[104X
    [4X[25Xgap>[125X [27XBasisVectors( B );[127X[104X
    [4X[28X[ [ 1, 2, 7 ], [ 0, 1, -9/4 ] ][128X[104X
  [4X[32X[104X
  
  [1X61.6-2 UnderlyingLeftModule[101X
  
  [33X[1;0Y[29X[2XUnderlyingLeftModule[102X( [3XB[103X ) [32X attribute[133X
  
  [33X[0;0YFor a basis [3XB[103X of a free left module [22XV[122X, [2XUnderlyingLeftModule[102X returns [22XV[122X.[133X
  
  [33X[0;0YThe 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 [2XBasis[102X ([14X61.5-2[114X)); furthermore, in
  some  cases  it  is  convenient to test membership of a vector in the module
  before computing coefficients w.r.t. a basis.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XB:= Basis( GF(2)^6 );;  UnderlyingLeftModule( B );[127X[104X
    [4X[28X( GF(2)^6 )[128X[104X
  [4X[32X[104X
  
  [1X61.6-3 Coefficients[101X
  
  [33X[1;0Y[29X[2XCoefficients[102X( [3XB[103X, [3Xv[103X ) [32X operation[133X
  
  [33X[0;0YLet [22XV[122X be the underlying left module of the basis [3XB[103X, and [3Xv[103X a vector such that
  the  family  of  [3Xv[103X  is  the  elements  family  of  the  family  of  [22XV[122X.  Then
  [10XCoefficients(  [3XB[103X[10X, [3Xv[103X[10X )[110X is the list of coefficients of [3Xv[103X w.r.t. [3XB[103X if [3Xv[103X lies in
  [22XV[122X, and [9Xfail[109X otherwise.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XV:= VectorSpace( Rationals, [ [ 1, 2, 7 ], [ 1/2, 1/3, 5 ] ] );;[127X[104X
    [4X[25Xgap>[125X [27XB:= Basis( V, [ [ 1, 2, 7 ], [ 0, 1, -9/4 ] ] );;[127X[104X
    [4X[25Xgap>[125X [27XCoefficients( B, [ 1/2, 1/3, 5 ] );[127X[104X
    [4X[28X[ 1/2, -2/3 ][128X[104X
    [4X[25Xgap>[125X [27XCoefficients( B, [ 1, 0, 0 ] );[127X[104X
    [4X[28Xfail[128X[104X
  [4X[32X[104X
  
  [1X61.6-4 LinearCombination[101X
  
  [33X[1;0Y[29X[2XLinearCombination[102X( [3XB[103X, [3Xcoeff[103X ) [32X operation[133X
  
  [33X[0;0YIf  [3XB[103X  is  a  basis  object  (see [2XIsBasis[102X ([14X61.5-1[114X)) or a homogeneous list of
  length  [22Xn[122X,  and  [3Xcoeff[103X is a row vector of the same length, [2XLinearCombination[102X
  returns the vector [22X∑_{i = 1}^n [3Xcoeff[103X[i] * [3XB[103X[i][122X.[133X
  
  [33X[0;0YPerhaps the most important usage is the case where [3XB[103X forms a basis.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XV:= VectorSpace( Rationals, [ [ 1, 2, 7 ], [ 1/2, 1/3, 5 ] ] );;[127X[104X
    [4X[25Xgap>[125X [27XB:= Basis( V, [ [ 1, 2, 7 ], [ 0, 1, -9/4 ] ] );;[127X[104X
    [4X[25Xgap>[125X [27XLinearCombination( B, [ 1/2, -2/3 ] );[127X[104X
    [4X[28X[ 1/2, 1/3, 5 ][128X[104X
  [4X[32X[104X
  
  [1X61.6-5 EnumeratorByBasis[101X
  
  [33X[1;0Y[29X[2XEnumeratorByBasis[102X( [3XB[103X ) [32X attribute[133X
  
  [33X[0;0YFor  a basis [3XB[103X of the free left [22XF[122X-module [22XV[122X of dimension [22Xn[122X, [10XEnumeratorByBasis[110X
  returns  an  enumerator  that  loops  over  the  elements  of  [22XV[122X  as  linear
  combinations  of  the  vectors of [3XB[103X with coefficients the row vectors in the
  full  row  space  (see [2XFullRowSpace[102X  ([14X61.9-4[114X)) of dimension [22Xn[122X over [22XF[122X, in the
  succession given by the default enumerator of this row space.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XV:= GF(2)^3;;[127X[104X
    [4X[25Xgap>[125X [27Xenum:= EnumeratorByBasis( CanonicalBasis( V ) );;[127X[104X
    [4X[25Xgap>[125X [27XPrint( enum{ [ 1 .. 4 ] }, "\n" );[127X[104X
    [4X[28X[ [ 0*Z(2), 0*Z(2), 0*Z(2) ], [ 0*Z(2), 0*Z(2), Z(2)^0 ], [128X[104X
    [4X[28X  [ 0*Z(2), Z(2)^0, 0*Z(2) ], [ 0*Z(2), Z(2)^0, Z(2)^0 ] ][128X[104X
    [4X[25Xgap>[125X [27XB:= Basis( V, [ [ 1, 1, 1 ], [ 1, 1, 0 ], [ 1, 0, 0 ] ] * Z(2) );;[127X[104X
    [4X[25Xgap>[125X [27Xenum:= EnumeratorByBasis( B );;[127X[104X
    [4X[25Xgap>[125X [27XPrint( enum{ [ 1 .. 4 ] }, "\n" );[127X[104X
    [4X[28X[ [ 0*Z(2), 0*Z(2), 0*Z(2) ], [ Z(2)^0, 0*Z(2), 0*Z(2) ], [128X[104X
    [4X[28X  [ Z(2)^0, Z(2)^0, 0*Z(2) ], [ 0*Z(2), Z(2)^0, 0*Z(2) ] ][128X[104X
  [4X[32X[104X
  
  [1X61.6-6 IteratorByBasis[101X
  
  [33X[1;0Y[29X[2XIteratorByBasis[102X( [3XB[103X ) [32X operation[133X
  
  [33X[0;0YFor  a  basis  [3XB[103X of the free left [22XF[122X-module [22XV[122X of dimension [22Xn[122X, [10XIteratorByBasis[110X
  returns an iterator that loops over the elements of [22XV[122X as linear combinations
  of  the vectors of [3XB[103X with coefficients the row vectors in the full row space
  (see [2XFullRowSpace[102X  ([14X61.9-4[114X))  of dimension [22Xn[122X over [22XF[122X, in the succession given
  by the default enumerator of this row space.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XV:= GF(2)^3;;[127X[104X
    [4X[25Xgap>[125X [27Xiter:= IteratorByBasis( CanonicalBasis( V ) );;[127X[104X
    [4X[25Xgap>[125X [27Xfor i in [ 1 .. 4 ] do Print( NextIterator( iter ), "\n" ); od;[127X[104X
    [4X[28X[ 0*Z(2), 0*Z(2), 0*Z(2) ][128X[104X
    [4X[28X[ 0*Z(2), 0*Z(2), Z(2)^0 ][128X[104X
    [4X[28X[ 0*Z(2), Z(2)^0, 0*Z(2) ][128X[104X
    [4X[28X[ 0*Z(2), Z(2)^0, Z(2)^0 ][128X[104X
    [4X[25Xgap>[125X [27XB:= Basis( V, [ [ 1, 1, 1 ], [ 1, 1, 0 ], [ 1, 0, 0 ] ] * Z(2) );;[127X[104X
    [4X[25Xgap>[125X [27Xiter:= IteratorByBasis( B );;[127X[104X
    [4X[25Xgap>[125X [27Xfor i in [ 1 .. 4 ] do Print( NextIterator( iter ), "\n" ); od;[127X[104X
    [4X[28X[ 0*Z(2), 0*Z(2), 0*Z(2) ][128X[104X
    [4X[28X[ Z(2)^0, 0*Z(2), 0*Z(2) ][128X[104X
    [4X[28X[ Z(2)^0, Z(2)^0, 0*Z(2) ][128X[104X
    [4X[28X[ 0*Z(2), Z(2)^0, 0*Z(2) ][128X[104X
  [4X[32X[104X
  
  
  [1X61.7 [33X[0;0YOperations for Special Kinds of Bases[133X[101X
  
  [1X61.7-1 IsCanonicalBasis[101X
  
  [33X[1;0Y[29X[2XIsCanonicalBasis[102X( [3XB[103X ) [32X property[133X
  
  [33X[0;0YIf  the  underlying  free  left module [22XV[122X of the basis [3XB[103X supports a canonical
  basis  (see [2XCanonicalBasis[102X ([14X61.5-3[114X)) then [2XIsCanonicalBasis[102X returns [9Xtrue[109X if [3XB[103X
  is equal to the canonical basis of [22XV[122X, and [9Xfalse[109X otherwise.[133X
  
  [1X61.7-2 IsIntegralBasis[101X
  
  [33X[1;0Y[29X[2XIsIntegralBasis[102X( [3XB[103X ) [32X property[133X
  
  [33X[0;0YLet  [3XB[103X  be an [22XS[122X-basis of a [13Xfield[113X [22XF[122X for a subfield [22XS[122X of [22XF[122X, and let [22XR[122X and [22XM[122X be
  the  rings  of  algebraic integers in [22XS[122X and [22XF[122X, respectively. [10XIsIntegralBasis[110X
  returns [9Xtrue[109X if [3XB[103X is also an [22XR[122X-basis of [22XM[122X, and [9Xfalse[109X otherwise.[133X
  
  [1X61.7-3 IsNormalBasis[101X
  
  [33X[1;0Y[29X[2XIsNormalBasis[102X( [3XB[103X ) [32X property[133X
  
  [33X[0;0YLet  [3XB[103X  be  an  [22XS[122X-basis  of  a  [13Xfield[113X [22XF[122X for a subfield [22XS[122X of [22XF[122X. [10XIsNormalBasis[110X
  returns  [9Xtrue[109X  if  [3XB[103X  is  invariant  under the Galois group (see [2XGaloisGroup[102X
  ([14X58.3-1[114X)) of the field extension [22XF / S[122X, and [9Xfalse[109X otherwise.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XB:= CanonicalBasis( GaussianRationals );[127X[104X
    [4X[28XCanonicalBasis( GaussianRationals )[128X[104X
    [4X[25Xgap>[125X [27XIsIntegralBasis( B );  IsNormalBasis( B );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  
  [1X61.8 [33X[0;0YMutable Bases[133X[101X
  
  [33X[0;0YIt  is  useful  to  have  a [13Xmutable basis[113X 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.[133X
  
  [33X[0;0YNote 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 [2XUnderlyingLeftModule[102X ([14X61.6-2[114X)).[133X
  
  [33X[0;0YSo  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.[133X
  
  [33X[0;0YSince  [10XImmutable[110X  produces  an immutable copy of any [5XGAP[105X 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  [13Xmutable[113X [5XGAP[105X objects, hence these objects are unable to store attribute
  values.[133X
  
  [33X[0;0YBasic   operations   for   immutable   bases  are  [2XNrBasisVectors[102X  ([14X61.8-3[114X),
  [2XIsContainedInSpan[102X   ([14X61.8-5[114X),   [2XCloseMutableBasis[102X  ([14X61.8-6[114X),  [2XImmutableBasis[102X
  ([14X61.8-4[114X),  [2XCoefficients[102X  ([14X61.6-3[114X),  and  [2XBasisVectors[102X  ([14X61.6-1[114X). [2XShallowCopy[102X
  ([14X12.7-1[114X)  for  a  mutable  basis returns a mutable plain list containing the
  current basis vectors.[133X
  
  [33X[0;0YSince  mutable  bases do not admit arbitrary changes of their lists of basis
  vectors,  a  mutable basis is [13Xnot[113X a list. It is, however, a collection, more
  precisely its family (see [14X13.1[114X) equals the family of its collection of basis
  vectors.[133X
  
  [33X[0;0YMutable bases can be constructed with [10XMutableBasis[110X.[133X
  
  [33X[0;0YSimilar  to  the situation with bases (cf. [14X61.5[114X), [5XGAP[105X supports the following
  three kinds of mutable bases.[133X
  
  [33X[0;0YThe  [13Xgeneric  method[113X  of  [10XMutableBasis[110X  returns  a mutable basis that simply
  stores an immutable basis; clearly one wants to avoid this whenever possible
  with reasonable effort.[133X
  
  [33X[0;0YThere  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 [14X61.11[114X)  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.[133X
  
  [33X[0;0YThere are mutable bases that use special information to perform their tasks;
  examples are mutable bases of Gaussian row and matrix spaces.[133X
  
  [1X61.8-1 IsMutableBasis[101X
  
  [33X[1;0Y[29X[2XIsMutableBasis[102X( [3XMB[103X ) [32X Category[133X
  
  [33X[0;0YEvery mutable basis lies in the category [10XIsMutableBasis[110X.[133X
  
  [1X61.8-2 MutableBasis[101X
  
  [33X[1;0Y[29X[2XMutableBasis[102X( [3XR[103X, [3Xvectors[103X[, [3Xzero[103X] ) [32X operation[133X
  
  [33X[0;0Y[10XMutableBasis[110X  returns a mutable basis for the [3XR[103X-free module generated by the
  vectors  in  the list [3Xvectors[103X. The optional argument [3Xzero[103X is the zero vector
  of the module; it must be given if [3Xvectors[103X is empty.[133X
  
  [33X[0;0Y[13XNote[113X  that  [3Xvectors[103X  will in general [13Xnot[113X be the basis vectors of the mutable
  basis![133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMB:= MutableBasis( Rationals, [ [ 1, 2, 3 ], [ 0, 1, 0 ] ] );[127X[104X
    [4X[28X<mutable basis over Rationals, 2 vectors>[128X[104X
  [4X[32X[104X
  
  [1X61.8-3 NrBasisVectors[101X
  
  [33X[1;0Y[29X[2XNrBasisVectors[102X( [3XMB[103X ) [32X operation[133X
  
  [33X[0;0YFor  a  mutable basis [3XMB[103X, [10XNrBasisVectors[110X returns the current number of basis
  vectors  of [3XMB[103X. Note that this operation is [13Xnot[113X an attribute, as it makes no
  sense  to store the value. [10XNrBasisVectors[110X is used mainly as an equivalent of
  [10XDimension[110X for the underlying left module in the case of immutable bases.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMB:= MutableBasis( Rationals, [ [ 1, 1], [ 2, 2 ] ] );;[127X[104X
    [4X[25Xgap>[125X [27XNrBasisVectors( MB );[127X[104X
    [4X[28X1[128X[104X
  [4X[32X[104X
  
  [1X61.8-4 ImmutableBasis[101X
  
  [33X[1;0Y[29X[2XImmutableBasis[102X( [3XMB[103X[, [3XV[103X] ) [32X operation[133X
  
  [33X[0;0Y[2XImmutableBasis[102X  returns the immutable basis [22XB[122X with the same basis vectors as
  in the mutable basis [3XMB[103X.[133X
  
  [33X[0;0YIf   the   second   argument   [3XV[103X   is   present  then  [3XV[103X  is  the  value  of
  [2XUnderlyingLeftModule[102X  ([14X61.6-2[114X)  for [22XB[122X. 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 [3XMB[103X, for example if it is
  an algebra.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMB:= MutableBasis( Rationals, [ [ 1, 1 ], [ 2, 2 ] ] );;[127X[104X
    [4X[25Xgap>[125X [27XB:= ImmutableBasis( MB );[127X[104X
    [4X[28XSemiEchelonBasis( <vector space of dimension 1 over Rationals>, [128X[104X
    [4X[28X[ [ 1, 1 ] ] )[128X[104X
    [4X[25Xgap>[125X [27XUnderlyingLeftModule( B );[127X[104X
    [4X[28X<vector space of dimension 1 over Rationals>[128X[104X
  [4X[32X[104X
  
  [1X61.8-5 IsContainedInSpan[101X
  
  [33X[1;0Y[29X[2XIsContainedInSpan[102X( [3XMB[103X, [3Xv[103X ) [32X operation[133X
  
  [33X[0;0YFor  a  mutable  basis  [3XMB[103X  over  the  coefficient  ring  [22XR[122X  and a vector [3Xv[103X,
  [10XIsContainedInSpan[110X  returns [9Xtrue[109X is [3Xv[103X lies in the [22XR[122X-span of the current basis
  vectors of [3XMB[103X, and [9Xfalse[109X otherwise.[133X
  
  [1X61.8-6 CloseMutableBasis[101X
  
  [33X[1;0Y[29X[2XCloseMutableBasis[102X( [3XMB[103X, [3Xv[103X ) [32X operation[133X
  
  [33X[0;0YFor  a  mutable  basis  [3XMB[103X  over  the  coefficient  ring  [22XR[122X  and a vector [3Xv[103X,
  [2XCloseMutableBasis[102X changes [3XMB[103X such that afterwards it describes the [22XR[122X-span of
  the former basis vectors together with [3Xv[103X.[133X
  
  [33X[0;0Y[13XNote[113X  that  if  [3Xv[103X  enlarges the dimension then this does in general [13Xnot[113X mean
  that  [3Xv[103X  is  simply  added  to  the  basis  vectors  of [3XMB[103X. Usually a linear
  combination  of  [3Xv[103X  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 [2XIsSemiEchelonized[102X ([14X61.9-7[114X)).[133X
  
  [33X[0;0Y[2XCloseMutableBasis[102X  returns [9Xfalse[109X if [3Xv[103X was already in the [22XR[122X-span described by
  [3XMB[103X, and [9Xtrue[109X if [3XMB[103X got extended.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMB:= MutableBasis( Rationals, [ [ 1, 1, 3 ], [ 2, 2, 1 ] ] );[127X[104X
    [4X[28X<mutable basis over Rationals, 2 vectors>[128X[104X
    [4X[25Xgap>[125X [27XIsContainedInSpan( MB, [ 1, 0, 0 ] );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XCloseMutableBasis( MB, [ 1, 0, 0 ] );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XMB;[127X[104X
    [4X[28X<mutable basis over Rationals, 3 vectors>[128X[104X
    [4X[25Xgap>[125X [27XIsContainedInSpan( MB, [ 1, 0, 0 ] );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XCloseMutableBasis( MB, [ 1, 0, 0 ] );[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  
  [1X61.9 [33X[0;0YRow and Matrix Spaces[133X[101X
  
  [1X61.9-1 IsRowSpace[101X
  
  [33X[1;0Y[29X[2XIsRowSpace[102X( [3XV[103X ) [32X filter[133X
  
  [33X[0;0YA  [13Xrow  space[113X  in  [5XGAP[105X  is  a vector space that consists of row vectors (see
  Chapter [14X23[114X).[133X
  
  [1X61.9-2 IsMatrixSpace[101X
  
  [33X[1;0Y[29X[2XIsMatrixSpace[102X( [3XV[103X ) [32X filter[133X
  
  [33X[0;0YA  [13Xmatrix  space[113X  in  [5XGAP[105X  is  a vector space that consists of matrices (see
  Chapter [14X24[114X).[133X
  
  [1X61.9-3 IsGaussianSpace[101X
  
  [33X[1;0Y[29X[2XIsGaussianSpace[102X( [3XV[103X ) [32X filter[133X
  
  [33X[0;0YThe  filter  [2XIsGaussianSpace[102X  (see [14X13.2[114X)  for  the row space (see [2XIsRowSpace[102X
  ([14X61.9-1[114X))  or  matrix  space  (see [2XIsMatrixSpace[102X  ([14X61.9-2[114X)) [3XV[103X over a field [22XF[122X
  indicates   that   the  entries  of  all  row  vectors  or  matrices  in  [3XV[103X,
  respectively,  are  all contained in [22XF[122X. In this case, [3XV[103X is called a [13XGaussian[113X
  vector  space.  Bases  for  Gaussian  spaces  can be computed using Gaussian
  elimination for a given list of vector space generators.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmats:= [ [[1,1],[2,2]], [[3,4],[0,1]] ];;[127X[104X
    [4X[25Xgap>[125X [27XV:= VectorSpace( Rationals, mats );;[127X[104X
    [4X[25Xgap>[125X [27XIsGaussianSpace( V );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xmats[1][1][1]:= E(4);;   # an element in an extension field[127X[104X
    [4X[25Xgap>[125X [27XV:= VectorSpace( Rationals, mats );;[127X[104X
    [4X[25Xgap>[125X [27XIsGaussianSpace( V );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XV:= VectorSpace( Field( Rationals, [ E(4) ] ), mats );;[127X[104X
    [4X[25Xgap>[125X [27XIsGaussianSpace( V );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X61.9-4 FullRowSpace[101X
  
  [33X[1;0Y[29X[2XFullRowSpace[102X( [3XF[103X, [3Xn[103X ) [32X function[133X
  [33X[1;0Y[29X[2X\^[102X( [3XF[103X, [3Xn[103X ) [32X method[133X
  
  [33X[0;0YFor a field [3XF[103X and a nonnegative integer [3Xn[103X, [2XFullRowSpace[102X returns the [3XF[103X-vector
  space  that consists of all row vectors (see [2XIsRowVector[102X ([14X23.1-1[114X)) of length
  [3Xn[103X with entries in [3XF[103X.[133X
  
  [33X[0;0YAn alternative to construct this vector space is via [3XF[103X[10X^[110X[3Xn[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XFullRowSpace( GF( 9 ), 3 );[127X[104X
    [4X[28X( GF(3^2)^3 )[128X[104X
    [4X[25Xgap>[125X [27XGF(9)^3;           # the same as above[127X[104X
    [4X[28X( GF(3^2)^3 )[128X[104X
  [4X[32X[104X
  
  [1X61.9-5 FullMatrixSpace[101X
  
  [33X[1;0Y[29X[2XFullMatrixSpace[102X( [3XF[103X, [3Xm[103X, [3Xn[103X ) [32X function[133X
  [33X[1;0Y[29X[2X\^[102X( [3XF[103X, [3Xdims[103X ) [32X method[133X
  
  [33X[0;0YFor a field [3XF[103X and two positive integers [3Xm[103X and [3Xn[103X, [2XFullMatrixSpace[102X returns the
  [3XF[103X-vector  space that consists of all [3Xm[103X by [3Xn[103X matrices (see [2XIsMatrix[102X ([14X24.2-1[114X))
  with entries in [3XF[103X.[133X
  
  [33X[0;0YIf  [3Xm[103X[10X  =  [110X[3Xn[103X  then  the  result  is in fact an algebra (see [2XFullMatrixAlgebra[102X
  ([14X62.5-4[114X)).[133X
  
  [33X[0;0YAn alternative to construct this vector space is via [3XF[103X[10X^[[110X[3Xm[103X,[3Xn[103X[10X][110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XFullMatrixSpace( GF(2), 4, 5 );[127X[104X
    [4X[28X( GF(2)^[ 4, 5 ] )[128X[104X
    [4X[25Xgap>[125X [27XGF(2)^[ 4, 5 ];    # the same as above[127X[104X
    [4X[28X( GF(2)^[ 4, 5 ] )[128X[104X
  [4X[32X[104X
  
  [1X61.9-6 DimensionOfVectors[101X
  
  [33X[1;0Y[29X[2XDimensionOfVectors[102X( [3XM[103X ) [32X attribute[133X
  
  [33X[0;0YFor a left module [3XM[103X that consists of row vectors (see [2XIsRowModule[102X ([14X57.3-6[114X)),
  [2XDimensionOfVectors[102X  returns the common length of all row vectors in [3XM[103X. For a
  left  module  [3XM[103X  that  consists  of  matrices (see [2XIsMatrixModule[102X ([14X57.3-7[114X)),
  [2XDimensionOfVectors[102X  returns  the common matrix dimensions (see [2XDimensionsMat[102X
  ([14X24.4-1[114X)) of all matrices in [3XM[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDimensionOfVectors( GF(2)^5 );[127X[104X
    [4X[28X5[128X[104X
    [4X[25Xgap>[125X [27XDimensionOfVectors( GF(2)^[2,3] );[127X[104X
    [4X[28X[ 2, 3 ][128X[104X
  [4X[32X[104X
  
  [1X61.9-7 IsSemiEchelonized[101X
  
  [33X[1;0Y[29X[2XIsSemiEchelonized[102X( [3XB[103X ) [32X property[133X
  
  [33X[0;0YLet  [3XB[103X  be  a basis of a Gaussian row or matrix space [22XV[122X (see [2XIsGaussianSpace[102X
  ([14X61.9-3[114X)) over the field [22XF[122X.[133X
  
  [33X[0;0YIf  [22XV[122X  is a row space then [3XB[103X 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 [22XF[122X, and all values exactly below these pivot elements are the
  zero of [22XF[122X (cf. [2XSemiEchelonMat[102X ([14X24.10-1[114X)).[133X
  
  [33X[0;0YIf  [22XV[122X is a matrix space then [3XB[103X is semi-echelonized if the matrix obtained by
  replacing   each   basis   vector  by  the  concatenation  of  its  rows  is
  semi-echelonized (see above, cf. [2XSemiEchelonMats[102X ([14X24.10-4[114X)).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XV:= GF(2)^2;;[127X[104X
    [4X[25Xgap>[125X [27XB1:= Basis( V, [ [ 0, 1 ], [ 1, 0 ] ] * Z(2) );;[127X[104X
    [4X[25Xgap>[125X [27XIsSemiEchelonized( B1 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XB2:= Basis( V, [ [ 0, 1 ], [ 1, 1 ] ] * Z(2) );;[127X[104X
    [4X[25Xgap>[125X [27XIsSemiEchelonized( B2 );[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X61.9-8 SemiEchelonBasis[101X
  
  [33X[1;0Y[29X[2XSemiEchelonBasis[102X( [3XV[103X[, [3Xvectors[103X] ) [32X attribute[133X
  [33X[1;0Y[29X[2XSemiEchelonBasisNC[102X( [3XV[103X, [3Xvectors[103X ) [32X operation[133X
  
  [33X[0;0YLet  [3XV[103X  be  a  Gaussian  row  or  matrix  vector  space  over  the  field  [22XF[122X
  (see [2XIsGaussianSpace[102X ([14X61.9-3[114X), [2XIsRowSpace[102X ([14X61.9-1[114X), [2XIsMatrixSpace[102X ([14X61.9-2[114X)).[133X
  
  [33X[0;0YCalled  with  [3XV[103X  as the only argument, [2XSemiEchelonBasis[102X returns a basis of [3XV[103X
  that has the property [2XIsSemiEchelonized[102X ([14X61.9-7[114X).[133X
  
  [33X[0;0YIf  additionally  a  list  [3Xvectors[103X  of  vectors  in  [3XV[103X is given that forms a
  semi-echelonized  basis  of  [3XV[103X  then [2XSemiEchelonBasis[102X returns this basis; if
  [3Xvectors[103X do not form a basis of [3XV[103X then [9Xfail[109X is returned.[133X
  
  [33X[0;0Y[2XSemiEchelonBasisNC[102X   does   the   same   as  the  two  argument  version  of
  [2XSemiEchelonBasis[102X,  except  that  it  is  not  checked whether [3Xvectors[103X form a
  semi-echelonized basis.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XV:= GF(2)^2;;[127X[104X
    [4X[25Xgap>[125X [27XB:= SemiEchelonBasis( V );[127X[104X
    [4X[28XSemiEchelonBasis( ( GF(2)^2 ), ... )[128X[104X
    [4X[25Xgap>[125X [27XPrint( BasisVectors( B ), "\n" );[127X[104X
    [4X[28X[ [ Z(2)^0, 0*Z(2) ], [ 0*Z(2), Z(2)^0 ] ][128X[104X
    [4X[25Xgap>[125X [27XB:= SemiEchelonBasis( V, [ [ 1, 1 ], [ 0, 1 ] ] * Z(2) );[127X[104X
    [4X[28XSemiEchelonBasis( ( GF(2)^2 ), <an immutable 2x2 matrix over GF2> )[128X[104X
    [4X[25Xgap>[125X [27XPrint( BasisVectors( B ), "\n" );[127X[104X
    [4X[28X[ [ Z(2)^0, Z(2)^0 ], [ 0*Z(2), Z(2)^0 ] ][128X[104X
    [4X[25Xgap>[125X [27XCoefficients( B, [ 0, 1 ] * Z(2) );[127X[104X
    [4X[28X[ 0*Z(2), Z(2)^0 ][128X[104X
    [4X[25Xgap>[125X [27XCoefficients( B, [ 1, 0 ] * Z(2) );[127X[104X
    [4X[28X[ Z(2)^0, Z(2)^0 ][128X[104X
    [4X[25Xgap>[125X [27XSemiEchelonBasis( V, [ [ 0, 1 ], [ 1, 1 ] ] * Z(2) );[127X[104X
    [4X[28Xfail[128X[104X
  [4X[32X[104X
  
  [1X61.9-9 IsCanonicalBasisFullRowModule[101X
  
  [33X[1;0Y[29X[2XIsCanonicalBasisFullRowModule[102X( [3XB[103X ) [32X property[133X
  
  [33X[0;0Y[2XIsCanonicalBasisFullRowModule[102X  returns  [9Xtrue[109X  if  [3XB[103X  is  the canonical basis
  (see [2XIsCanonicalBasis[102X  ([14X61.7-1[114X))  of  a full row module (see [2XIsFullRowModule[102X
  ([14X57.3-8[114X)), and [9Xfalse[109X otherwise.[133X
  
  [33X[0;0YThe  [13Xcanonical  basis[113X  of  a  Gaussian  row  space  is defined as the unique
  semi-echelonized  (see [2XIsSemiEchelonized[102X ([14X61.9-7[114X)) basis with the additional
  property  that  for  [22Xj > i[122X the position of the pivot of row [22Xj[122X is bigger than
  the  position  of  the  pivot  of row [22Xi[122X, and that each pivot column contains
  exactly one nonzero entry.[133X
  
  [1X61.9-10 IsCanonicalBasisFullMatrixModule[101X
  
  [33X[1;0Y[29X[2XIsCanonicalBasisFullMatrixModule[102X( [3XB[103X ) [32X property[133X
  
  [33X[0;0Y[2XIsCanonicalBasisFullMatrixModule[102X  returns  [9Xtrue[109X  if [3XB[103X is the canonical basis
  (see [2XIsCanonicalBasis[102X     ([14X61.7-1[114X))     of     a    full    matrix    module
  (see [2XIsFullMatrixModule[102X ([14X57.3-10[114X)), and [9Xfalse[109X otherwise.[133X
  
  [33X[0;0YThe  [13Xcanonical  basis[113X  of  a  Gaussian matrix space is defined as the unique
  semi-echelonized  (see [2XIsSemiEchelonized[102X  ([14X61.9-7[114X)) basis for which the list
  of  concatenations  of  the  basis  vectors forms the canonical basis of the
  corresponding Gaussian row space.[133X
  
  [1X61.9-11 NormedRowVectors[101X
  
  [33X[1;0Y[29X[2XNormedRowVectors[102X( [3XV[103X ) [32X attribute[133X
  
  [33X[0;0YFor  a finite Gaussian row space [3XV[103X (see [2XIsRowSpace[102X ([14X61.9-1[114X), [2XIsGaussianSpace[102X
  ([14X61.9-3[114X)),  [2XNormedRowVectors[102X  returns  a  list of those nonzero vectors in [3XV[103X
  that have a one in the first nonzero component.[133X
  
  [33X[0;0YThe  result  list  can  be  used as action domain for the action of a matrix
  group   via   [2XOnLines[102X   ([14X41.2-12[114X),   which  yields  the  natural  action  on
  one-dimensional subspaces of [3XV[103X (see also [2XSubspaces[102X ([14X61.4-1[114X)).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xvecs:= NormedRowVectors( GF(3)^2 );[127X[104X
    [4X[28X[ [ 0*Z(3), Z(3)^0 ], [ Z(3)^0, 0*Z(3) ], [ Z(3)^0, Z(3)^0 ], [128X[104X
    [4X[28X  [ Z(3)^0, Z(3) ] ][128X[104X
    [4X[25Xgap>[125X [27XAction( GL(2,3), vecs, OnLines );[127X[104X
    [4X[28XGroup([ (3,4), (1,2,4) ])[128X[104X
  [4X[32X[104X
  
  [1X61.9-12 SiftedVector[101X
  
  [33X[1;0Y[29X[2XSiftedVector[102X( [3XB[103X, [3Xv[103X ) [32X operation[133X
  
  [33X[0;0YLet  [3XB[103X  be  a  semi-echelonized  basis (see [2XIsSemiEchelonized[102X ([14X61.9-7[114X)) of a
  Gaussian  row  or matrix space [22XV[122X (see [2XIsGaussianSpace[102X ([14X61.9-3[114X)), and [3Xv[103X a row
  vector  or matrix, respectively, of the same dimension as the elements in [22XV[122X.
  [10XSiftedVector[110X  returns the [13Xresiduum[113X of [3Xv[103X with respect to [3XB[103X, which is obtained
  by  successively  cleaning the pivot positions in [3Xv[103X by subtracting multiples
  of the basis vectors in [3XB[103X. So the result is the zero vector in [22XV[122X if and only
  if [3Xv[103X lies in [22XV[122X.[133X
  
  [33X[0;0Y[3XB[103X may also be a mutable basis (see [14X61.8[114X) of a Gaussian row or matrix space.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XV:= VectorSpace( Rationals, [ [ 1, 2, 7 ], [ 1/2, 1/3, 5 ] ] );;[127X[104X
    [4X[25Xgap>[125X [27XB:= Basis( V );;[127X[104X
    [4X[25Xgap>[125X [27XSiftedVector( B, [ 1, 2, 8 ] );[127X[104X
    [4X[28X[ 0, 0, 1 ][128X[104X
  [4X[32X[104X
  
  
  [1X61.10 [33X[0;0YVector Space Homomorphisms[133X[101X
  
  [33X[0;0Y[13XVector   space   homomorphisms[113X   (or   [13Xlinear   mappings[113X)   are  defined  in
  Section [14X32.11[114X.  [5XGAP[105X  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.[133X
  
  [33X[0;0Y[22XF[122X-linear  mappings  with same source and same range can be added, so one can
  form vector spaces of linear mappings.[133X
  
  [1X61.10-1 LeftModuleGeneralMappingByImages[101X
  
  [33X[1;0Y[29X[2XLeftModuleGeneralMappingByImages[102X( [3XV[103X, [3XW[103X, [3Xgens[103X, [3Ximgs[103X ) [32X operation[133X
  
  [33X[0;0YLet  [3XV[103X and [3XW[103X be two left modules over the same left acting domain [22XR[122X and [3Xgens[103X
  and  [3Ximgs[103X  lists  (of the same length) of elements in [3XV[103X and [3XW[103X, respectively.
  [2XLeftModuleGeneralMappingByImages[102X  returns  the general mapping with source [3XV[103X
  and  range  [3XW[103X  that  is  defined  by  mapping  the  elements  in [3Xgens[103X to the
  corresponding elements in [3Ximgs[103X, and taking the [22XR[122X-linear closure.[133X
  
  [33X[0;0Y[3Xgens[103X  need  not generate [3XV[103X as a left [22XR[122X-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 [2XIsMapping[102X ([14X32.3-3[114X)).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XV:= Rationals^2;;[127X[104X
    [4X[25Xgap>[125X [27XW:= VectorSpace( Rationals, [ [1,2,3], [1,0,1] ] );;[127X[104X
    [4X[25Xgap>[125X [27Xf:= LeftModuleGeneralMappingByImages( V, W,[127X[104X
    [4X[25X>[125X [27X                               [[1,0],[2,0]], [[1,0,1],[1,0,1] ] );[127X[104X
    [4X[28X[ [ 1, 0 ], [ 2, 0 ] ] -> [ [ 1, 0, 1 ], [ 1, 0, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27XIsMapping( f );[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X61.10-2 LeftModuleHomomorphismByImages[101X
  
  [33X[1;0Y[29X[2XLeftModuleHomomorphismByImages[102X( [3XV[103X, [3XW[103X, [3Xgens[103X, [3Ximgs[103X ) [32X function[133X
  [33X[1;0Y[29X[2XLeftModuleHomomorphismByImagesNC[102X( [3XV[103X, [3XW[103X, [3Xgens[103X, [3Ximgs[103X ) [32X operation[133X
  
  [33X[0;0YLet  [3XV[103X and [3XW[103X be two left modules over the same left acting domain [22XR[122X and [3Xgens[103X
  and  [3Ximgs[103X  lists  (of the same length) of elements in [3XV[103X and [3XW[103X, respectively.
  [2XLeftModuleHomomorphismByImages[102X  returns  the left [22XR[122X-module homomorphism with
  source  [3XV[103X and range [3XW[103X that is defined by mapping the elements in [3Xgens[103X to the
  corresponding elements in [3Ximgs[103X.[133X
  
  [33X[0;0YIf  [3Xgens[103X does not generate [3XV[103X or if the homomorphism does not exist (i.e., if
  mapping  the  generators describes only a multi-valued mapping) then [9Xfail[109X is
  returned.  For  creating  a  possibly  multi-valued mapping from [3XV[103X to [3XW[103X that
  respects    addition,    multiplication,    and    scalar    multiplication,
  [2XLeftModuleGeneralMappingByImages[102X ([14X61.10-1[114X) can be used.[133X
  
  [33X[0;0Y[2XLeftModuleHomomorphismByImagesNC[102X        does        the        same       as
  [2XLeftModuleHomomorphismByImages[102X, except that it omits all checks.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XV:=Rationals^2;;[127X[104X
    [4X[25Xgap>[125X [27XW:=VectorSpace( Rationals, [ [ 1, 0, 1 ], [ 1, 2, 3 ] ] );;[127X[104X
    [4X[25Xgap>[125X [27Xf:=LeftModuleHomomorphismByImages( V, W,[127X[104X
    [4X[25X>[125X [27X[ [ 1, 0 ], [ 0, 1 ] ], [ [ 1, 0, 1 ], [ 1, 2, 3 ] ] );[127X[104X
    [4X[28X[ [ 1, 0 ], [ 0, 1 ] ] -> [ [ 1, 0, 1 ], [ 1, 2, 3 ] ][128X[104X
    [4X[25Xgap>[125X [27XImage( f, [1,1] );[127X[104X
    [4X[28X[ 2, 2, 4 ][128X[104X
  [4X[32X[104X
  
  [1X61.10-3 LeftModuleHomomorphismByMatrix[101X
  
  [33X[1;0Y[29X[2XLeftModuleHomomorphismByMatrix[102X( [3XBS[103X, [3Xmatrix[103X, [3XBR[103X ) [32X operation[133X
  
  [33X[0;0YLet  [3XBS[103X  and  [3XBR[103X  be  bases  of  the  left  [22XR[122X-modules [22XV[122X and [22XW[122X, respectively.
  [2XLeftModuleHomomorphismByMatrix[102X returns the [22XR[122X-linear mapping from [22XV[122X to [22XW[122X that
  is  defined  by  the  matrix [3Xmatrix[103X, as follows. The image of the [22Xi[122X-th basis
  vector  of  [3XBS[103X  is  the  linear  combination of the basis vectors of [3XBR[103X with
  coefficients the [22Xi[122X-th row of [3Xmatrix[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XV:= Rationals^2;;[127X[104X
    [4X[25Xgap>[125X [27XW:= VectorSpace( Rationals, [ [ 1, 0, 1 ], [ 1, 2, 3 ] ] );;[127X[104X
    [4X[25Xgap>[125X [27Xf:= LeftModuleHomomorphismByMatrix( Basis( V ),[127X[104X
    [4X[25X>[125X [27X[ [ 1, 2 ], [ 3, 1 ] ], Basis( W ) );[127X[104X
    [4X[28X<linear mapping by matrix, ( Rationals^[128X[104X
    [4X[28X2 ) -> <vector space over Rationals, with 2 generators>>[128X[104X
  [4X[32X[104X
  
  [1X61.10-4 NaturalHomomorphismBySubspace[101X
  
  [33X[1;0Y[29X[2XNaturalHomomorphismBySubspace[102X( [3XV[103X, [3XW[103X ) [32X operation[133X
  
  [33X[0;0YFor an [22XR[122X-vector space [3XV[103X and a subspace [3XW[103X of [3XV[103X, [2XNaturalHomomorphismBySubspace[102X
  returns  the  [22XR[122X-linear  mapping that is the natural projection of [3XV[103X onto the
  factor space [10X[3XV[103X[10X / [3XW[103X[10X[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XV:= Rationals^3;;[127X[104X
    [4X[25Xgap>[125X [27XW:= VectorSpace( Rationals, [ [ 1, 1, 1 ] ] );;[127X[104X
    [4X[25Xgap>[125X [27Xf:= NaturalHomomorphismBySubspace( V, W );[127X[104X
    [4X[28X<linear mapping by matrix, ( Rationals^3 ) -> ( Rationals^2 )>[128X[104X
  [4X[32X[104X
  
  [1X61.10-5 Hom[101X
  
  [33X[1;0Y[29X[2XHom[102X( [3XF[103X, [3XV[103X, [3XW[103X ) [32X operation[133X
  
  [33X[0;0YFor  a  field  [3XF[103X  and  two  vector  spaces  [3XV[103X  and [3XW[103X that can be regarded as
  [3XF[103X-modules (see [2XAsLeftModule[102X ([14X57.1-5[114X)), [2XHom[102X returns the [3XF[103X-vector space of all
  [3XF[103X-linear mappings from [3XV[103X to [3XW[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XV:= Rationals^2;;[127X[104X
    [4X[25Xgap>[125X [27XW:= VectorSpace( Rationals, [ [ 1, 0, 1 ], [ 1, 2, 3 ] ] );;[127X[104X
    [4X[25Xgap>[125X [27XH:= Hom( Rationals, V, W );[127X[104X
    [4X[28XHom( Rationals, ( Rationals^2 ), <vector space over Rationals, with [128X[104X
    [4X[28X2 generators> )[128X[104X
    [4X[25Xgap>[125X [27XDimension( H );[127X[104X
    [4X[28X4[128X[104X
  [4X[32X[104X
  
  [1X61.10-6 End[101X
  
  [33X[1;0Y[29X[2XEnd[102X( [3XF[103X, [3XV[103X ) [32X operation[133X
  
  [33X[0;0YFor  a  field  [3XF[103X  and  a  vector space [3XV[103X that can be regarded as an [3XF[103X-module
  (see [2XAsLeftModule[102X  ([14X57.1-5[114X)),  [2XEnd[102X  returns  the  [3XF[103X-algebra  of all [3XF[103X-linear
  mappings from [3XV[103X to [3XV[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XA:= End( Rationals, Rationals^2 );[127X[104X
    [4X[28XEnd( Rationals, ( Rationals^2 ) )[128X[104X
    [4X[25Xgap>[125X [27XDimension( A );[127X[104X
    [4X[28X4[128X[104X
  [4X[32X[104X
  
  [1X61.10-7 IsFullHomModule[101X
  
  [33X[1;0Y[29X[2XIsFullHomModule[102X( [3XM[103X ) [32X property[133X
  
  [33X[0;0YA  [13Xfull  hom  module[113X  is  a module of all [22XR[122X-linear mappings between two left
  [22XR[122X-modules.  The  function  [2XHom[102X ([14X61.10-5[114X) can be used to construct a full hom
  module.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XV:= Rationals^2;;[127X[104X
    [4X[25Xgap>[125X [27XW:= VectorSpace( Rationals, [ [ 1, 0, 1 ], [ 1, 2, 3 ] ] );;[127X[104X
    [4X[25Xgap>[125X [27XH:= Hom( Rationals, V, W );;[127X[104X
    [4X[25Xgap>[125X [27XIsFullHomModule( H );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X61.10-8 IsPseudoCanonicalBasisFullHomModule[101X
  
  [33X[1;0Y[29X[2XIsPseudoCanonicalBasisFullHomModule[102X( [3XB[103X ) [32X property[133X
  
  [33X[0;0YA  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.[133X
  
  [33X[0;0YNote  that  this  is  not  a  canonical  basis (see [2XCanonicalBasis[102X ([14X61.5-3[114X))
  because it depends on the stored bases of source and range.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsPseudoCanonicalBasisFullHomModule( Basis( H ) );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X61.10-9 IsLinearMappingsModule[101X
  
  [33X[1;0Y[29X[2XIsLinearMappingsModule[102X( [3XV[103X ) [32X filter[133X
  
  [33X[0;0YIf  an  [22XF[122X-vector  space  [3XV[103X is in the filter [2XIsLinearMappingsModule[102X then this
  expresses  that [3XV[103X consists of linear mappings, and that [3XV[103X is handled via the
  mechanism  of  nice  bases (see [14X61.11[114X), in the following way. Let [22XS[122X and [22XR[122X be
  the  source  and  the  range,  respectively,  of each mapping in [22XV[122X. Then the
  [2XNiceFreeLeftModuleInfo[102X  ([14X61.11-3[114X) value of [3XV[103X is a record with the components
  [10Xbasissource[110X  (a  basis  [22XB_S[122X of [22XS[122X) and [10Xbasisrange[110X (a basis [22XB_R[122X of [22XR[122X), and the
  [2XNiceVector[102X ([14X61.11-2[114X) value of [22Xv ∈ [3XV[103X[122X is defined as the matrix of the [22XF[122X-linear
  mapping [22Xv[122X w.r.t. the bases [22XB_S[122X and [22XB_R[122X.[133X
  
  
  [1X61.11 [33X[0;0YVector Spaces Handled By Nice Bases[133X[101X
  
  [33X[0;0YThere  are  kinds  of  free  [22XR[122X-modules  for which efficient computations are
  possible  because  the  elements are [21Xnice[121X, for example subspaces of full row
  modules or of full matrix modules. In other cases, a [21Xnice[121X canonical basis is
  known  that allows one to do the necessary computations in the corresponding
  row module, for example algebras given by structure constants.[133X
  
  [33X[0;0YIn  many  other situations, one knows at least an isomorphism from the given
  module [22XV[122X to a [21Xnicer[121X free left module [22XW[122X, in the sense that for each vector in
  [22XV[122X, the image in [22XW[122X can easily be computed, and analogously for each vector in
  [22XW[122X, one can compute the preimage in [22XV[122X.[133X
  
  [33X[0;0YThis  allows  one  to  delegate  computations  w.r.t. a  basis [22XB[122X of [22XV[122X to the
  corresponding  basis [22XC[122X of [22XW[122X. We call [22XW[122X the [13Xnice free left module[113X of [22XV[122X, and [22XC[122X
  the  [13Xnice  basis[113X  of  [22XB[122X.  (Note  that  it  may  happen that also [22XC[122X delegates
  questions to a [21Xnicer[121X basis.) The basis [22XB[122X indicates the intended behaviour by
  the  filter  [2XIsBasisByNiceBasis[102X  ([14X61.11-5[114X),  and  stores  [22XC[122X  as value of the
  attribute  [2XNiceBasis[102X  ([14X61.11-4[114X).  [22XV[122X  indicates the intended behaviour by the
  filter  [2XIsHandledByNiceBasis[102X  ([14X61.11-6[114X),  and  stores  [22XW[122X  as  value  of  the
  attribute [2XNiceFreeLeftModule[102X ([14X61.11-1[114X).[133X
  
  [33X[0;0YThe  bijection  between  [22XV[122X  and [22XW[122X is implemented by the functions [2XNiceVector[102X
  ([14X61.11-2[114X) and [2XUglyVector[102X ([14X61.11-2[114X); additional data needed to compute images
  and preimages can be stored as value of [2XNiceFreeLeftModuleInfo[102X ([14X61.11-3[114X).[133X
  
  [1X61.11-1 NiceFreeLeftModule[101X
  
  [33X[1;0Y[29X[2XNiceFreeLeftModule[102X( [3XV[103X ) [32X attribute[133X
  
  [33X[0;0YFor  a  free  left module [3XV[103X that is handled via the mechanism of nice bases,
  this attribute stores the associated free left module to which the tasks are
  delegated.[133X
  
  [1X61.11-2 NiceVector[101X
  
  [33X[1;0Y[29X[2XNiceVector[102X( [3XV[103X, [3Xv[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XUglyVector[102X( [3XV[103X, [3Xr[103X ) [32X operation[133X
  
  [33X[0;0Y[2XNiceVector[102X and [2XUglyVector[102X provide the linear bijection between the free left
  module [3XV[103X and [10X[3XW[103X[10X:= NiceFreeLeftModule( [3XV[103X[10X )[110X.[133X
  
  [33X[0;0YIf  [3Xv[103X lies in the elements family of the family of [3XV[103X then [10XNiceVector( [3Xv[103X[10X )[110X is
  either [9Xfail[109X or an element in the elements family of the family of [3XW[103X.[133X
  
  [33X[0;0YIf  [3Xr[103X lies in the elements family of the family of [3XW[103X then [10XUglyVector( [3Xr[103X[10X )[110X is
  either [9Xfail[109X or an element in the elements family of the family of [3XV[103X.[133X
  
  [33X[0;0YIf  [3Xv[103X  lies  in  [3XV[103X  (which  usually  [13Xcannot[113X be checked without using [3XW[103X) then
  [10XUglyVector(  [3XV[103X[10X,  NiceVector( [3XV[103X[10X, [3Xv[103X[10X ) ) = [3Xv[103X[10X[110X. If [3Xr[103X lies in [3XW[103X (which usually [13Xcan[113X
  be checked) then [10XNiceVector( [3XV[103X[10X, UglyVector( [3XV[103X[10X, [3Xr[103X[10X ) ) = [3Xr[103X[10X[110X.[133X
  
  [33X[0;0Y(This  allows one to implement for example a membership test for [3XV[103X using the
  membership test in [3XW[103X.)[133X
  
  [1X61.11-3 NiceFreeLeftModuleInfo[101X
  
  [33X[1;0Y[29X[2XNiceFreeLeftModuleInfo[102X( [3XV[103X ) [32X attribute[133X
  
  [33X[0;0YFor  a  free  left module [3XV[103X that is handled via the mechanism of nice bases,
  this  operation  has to provide the necessary information (if any) for calls
  of [2XNiceVector[102X ([14X61.11-2[114X) and [2XUglyVector[102X ([14X61.11-2[114X).[133X
  
  [1X61.11-4 NiceBasis[101X
  
  [33X[1;0Y[29X[2XNiceBasis[102X( [3XB[103X ) [32X attribute[133X
  
  [33X[0;0YLet  [3XB[103X be a basis of a free left module [3XV[103X that is handled via nice bases. If
  [3XB[103X  has  no  basis  vectors stored at the time of the first call to [10XNiceBasis[110X
  then  [10XNiceBasis(  [3XB[103X[10X  )[110X  is  obtained as [10XBasis( NiceFreeLeftModule( [3XV[103X[10X ) )[110X. If
  basis  vectors  are  stored then [10XNiceBasis( [3XB[103X[10X )[110X is the result of the call of
  [10XBasis[110X  with  arguments  [10XNiceFreeLeftModule( [3XV[103X[10X )[110X and the [10XNiceVector[110X values of
  the basis vectors of [3XB[103X.[133X
  
  [33X[0;0YNote  that  the  result is [9Xfail[109X if and only if the [21Xbasis vectors[121X stored in [3XB[103X
  are in fact not basis vectors.[133X
  
  [33X[0;0YThe  attributes  [10XGeneratorsOfLeftModule[110X  of the underlying left modules of [3XB[103X
  and  the  result  of  [10XNiceBasis[110X  correspond  via  [2XNiceVector[102X  ([14X61.11-2[114X)  and
  [2XUglyVector[102X ([14X61.11-2[114X).[133X
  
  [1X61.11-5 IsBasisByNiceBasis[101X
  
  [33X[1;0Y[29X[2XIsBasisByNiceBasis[102X( [3XB[103X ) [32X Category[133X
  
  [33X[0;0YThis  filter  indicates  that  the  basis  [3XB[103X  delegates  tasks  such  as the
  computation  of  coefficients  (see [2XCoefficients[102X  ([14X61.6-3[114X)) to a basis of an
  isomorphic [21Xnicer[121X free left module.[133X
  
  [1X61.11-6 IsHandledByNiceBasis[101X
  
  [33X[1;0Y[29X[2XIsHandledByNiceBasis[102X( [3XM[103X ) [32X Category[133X
  
  [33X[0;0YFor  a  free  left module [3XM[103X in this category, essentially all operations are
  performed using a [21Xnicer[121X free left module, which is usually a row module.[133X
  
  
  [1X61.12 [33X[0;0YHow to Implement New Kinds of Vector Spaces[133X[101X
  
  [1X61.12-1 DeclareHandlingByNiceBasis[101X
  
  [33X[1;0Y[29X[2XDeclareHandlingByNiceBasis[102X( [3Xname[103X, [3Xinfo[103X ) [32X function[133X
  [33X[1;0Y[29X[2XInstallHandlingByNiceBasis[102X( [3Xname[103X, [3Xrecord[103X ) [32X function[133X
  
  [33X[0;0YThese  functions  are used to implement a new kind of free left modules that
  shall be handled via the mechanism of nice bases (see [14X61.11[114X).[133X
  
  [33X[0;0Y[3Xname[103X  must  be a string, a filter [22Xf[122X with this name is created, and a logical
  implication from [22Xf[122X to [2XIsHandledByNiceBasis[102X ([14X61.11-6[114X) is installed.[133X
  
  [33X[0;0Y[3Xrecord[103X must be a record with the following components.[133X
  
  [8X[10Xdetect[110X[8X [108X
        [33X[0;6Ya  function  of  four arguments [22XR[122X, [22Xl[122X, [22XV[122X, and [22Xz[122X, where [22XV[122X is a free left
        module over the ring [22XR[122X with generators the list or collection [22Xl[122X, and [22Xz[122X
        is  either  the  zero  element of [22XV[122X or [9Xfalse[109X (then [22Xl[122X is nonempty); the
        function  returns  [9Xtrue[109X  if  [22XV[122X  shall  lie  in the filter [22Xf[122X, and [9Xfalse[109X
        otherwise;  the  return value may also be [9Xfail[109X, which indicates that [22XV[122X
        is [13Xnot[113X to be handled via the mechanism of nice bases at all,[133X
  
  [8X[10XNiceFreeLeftModuleInfo[110X[8X [108X
        [33X[0;6Ythe [10XNiceFreeLeftModuleInfo[110X method for left modules in [22Xf[122X,[133X
  
  [8X[10XNiceVector[110X[8X [108X
        [33X[0;6Ythe  [10XNiceVector[110X  method  for  left modules [22XV[122X in [22Xf[122X; called with [22XV[122X and a
        vector  [22Xv ∈ V[122X, this function returns the nice vector [22Xr[122X associated with
        [22Xv[122X, and[133X
  
  [8X[10XUglyVector[110X[8X[108X
        [33X[0;6Ythe [2XUglyVector[102X ([14X61.11-2[114X) method for left modules [22XV[122X in [22Xf[122X; called with [22XV[122X
        and  a  vector  [22Xr[122X  in the [10XNiceFreeLeftModule[110X value of [22XV[122X, this function
        returns the vector [22Xv ∈ V[122X to which [22Xr[122X is associated.[133X
  
  [33X[0;0YThe  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
  [10XDeclareHandlingByNiceBasis[110X  and [10XInstallHandlingByNiceBasis[110X, which causes the
  installation  of  the necessary methods and adds the pair [22X[ f,[122X[10X[3Xrecord[103X[10X.detect[110X[22X][122X
  to   the   global   list  [10XNiceBasisFiltersInfo[110X.  The  [2XLeftModuleByGenerators[102X
  ([14X57.1-10[114X) methods call [2XCheckForHandlingByNiceBasis[102X ([14X61.12-3[114X), which sets the
  appropriate filter for the desired left module if applicable.[133X
  
  [1X61.12-2 NiceBasisFiltersInfo[101X
  
  [33X[1;0Y[29X[2XNiceBasisFiltersInfo[102X [32X global variable[133X
  
  [33X[0;0YAn overview of all kinds of vector spaces that are currently handled by nice
  bases  is  given  by  the global list [10XNiceBasisFiltersInfo[110X. Examples of such
  vector  spaces  are  vector  spaces  of  field  elements (but not the fields
  themselves)  and  non-Gaussian  row  and  matrix spaces (see [2XIsGaussianSpace[102X
  ([14X61.9-3[114X)).[133X
  
  [1X61.12-3 CheckForHandlingByNiceBasis[101X
  
  [33X[1;0Y[29X[2XCheckForHandlingByNiceBasis[102X( [3XR[103X, [3Xgens[103X, [3XM[103X, [3Xzero[103X ) [32X function[133X
  
  [33X[0;0YWhenever   a   free   left  module  is  constructed  for  which  the  filter
  [10XIsHandledByNiceBasis[110X  may  be  useful, [10XCheckForHandlingByNiceBasis[110X should be
  called.   (This   is   done  in  the  methods  for  [10XVectorSpaceByGenerators[110X,
  [10XAlgebraByGenerators[110X, [10XIdealByGenerators[110X etc. in the [5XGAP[105X library.)[133X
  
  [33X[0;0YThe  arguments of this function are the coefficient ring [3XR[103X, the list [3Xgens[103X of
  generators, the constructed module [3XM[103X itself, and the zero element [3Xzero[103X of [3XM[103X;
  if [3Xgens[103X is nonempty then the [3Xzero[103X value may also be [9Xfalse[109X.[133X
  
  
  [1X61.13 [33X[0;0YTensor Products and Exterior and Symmetric Powers[133X[101X
  
  [1X61.13-1 TensorProduct[101X
  
  [33X[1;0Y[29X[2XTensorProduct[102X( [3Xlist[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XTensorProduct[102X( [3XV[103X, [3XW[103X, [3X...[103X ) [32X operation[133X
  
  [33X[0;0YHere  [3Xlist[103X 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.[133X
  
  [33X[0;0YIn the second form, the vector spaces are given individually.[133X
  
  [33X[0;0YElements of the tensor product [22XV_1⊗ ⋯ ⊗ V_k[122X are linear combinations of [22Xv_1⊗⋯
  ⊗  v_k[122X,  where  the [22Xv_i[122X are arbitrary basis elements of [22XV_i[122X. In [5XGAP[105X a tensor
  element like that is printed as[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X   v_1<x> ... <x>v_k[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFurthermore, the zero of a tensor product is printed as[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X <0-tensor>[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThis  does  not  mean  that  all tensor products have the same zero element:
  zeros of different tensor products have different families.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XV:=TensorProduct(Rationals^2, Rationals^3);[127X[104X
    [4X[28X<vector space over Rationals, with 6 generators>[128X[104X
    [4X[25Xgap>[125X [27XBasis(V);[127X[104X
    [4X[28XBasis( <vector space over Rationals, with 6 generators>,[128X[104X
    [4X[28X[ 1*([ 0, 1 ]<x>[ 0, 0, 1 ]), 1*([ 0, 1 ]<x>[ 0, 1, 0 ]), [128X[104X
    [4X[28X  1*([ 0, 1 ]<x>[ 1, 0, 0 ]), 1*([ 1, 0 ]<x>[ 0, 0, 1 ]), [128X[104X
    [4X[28X  1*([ 1, 0 ]<x>[ 0, 1, 0 ]), 1*([ 1, 0 ]<x>[ 1, 0, 0 ]) ] )[128X[104X
  [4X[32X[104X
  
  [33X[0;0YSee also [2XKroneckerProduct[102X ([14X24.5-8[114X).[133X
  
  [1X61.13-2 ExteriorPower[101X
  
  [33X[1;0Y[29X[2XExteriorPower[102X( [3XV[103X, [3Xk[103X ) [32X operation[133X
  
  [33X[0;0YHere [3XV[103X must be a vector space. This function returns the [3Xk[103X-th exterior power
  of [3XV[103X.[133X
  
  [33X[0;0YElements  of  the  exterior power [22X⋀^k V[122X are linear combinations of [22Xv_i_1∧⋯ ∧
  v_i_k[122X,  where  the [22Xv_i_j[122X are basis elements of [22XV[122X, and [22X1 ≤ i_1 < i_2 ⋯ < i_k[122X.
  In [5XGAP[105X a wedge element like that is printed as[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X   v_1/\ ... /\v_k[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFurthermore, the zero of an exterior power is printed as[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X <0-wedge>[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThis  does  not  mean  that  all exterior powers have the same zero element:
  zeros of different exterior powers have different families.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XV:=ExteriorPower(Rationals^3, 2);[127X[104X
    [4X[28X<vector space of dimension 3 over Rationals>[128X[104X
    [4X[25Xgap>[125X [27XBasis(V);[127X[104X
    [4X[28XBasis( <vector space of dimension 3 over Rationals>, [ [128X[104X
    [4X[28X  1*([ 0, 1, 0 ]/\[ 0, 0, 1 ]), 1*([ 1, 0, 0 ]/\[ 0, 0, 1 ]), [128X[104X
    [4X[28X  1*([ 1, 0, 0 ]/\[ 0, 1, 0 ]) ] )[128X[104X
  [4X[32X[104X
  
  [1X61.13-3 SymmetricPower[101X
  
  [33X[1;0Y[29X[2XSymmetricPower[102X( [3XV[103X, [3Xk[103X ) [32X operation[133X
  
  [33X[0;0YHere  [3XV[103X  must  be  a  vector space. This function returns the [3Xk[103X-th symmetric
  power of [3XV[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XV:=SymmetricPower(Rationals^3, 2);[127X[104X
    [4X[28X<vector space over Rationals, with 6 generators>[128X[104X
    [4X[25Xgap>[125X [27XBasis(V);[127X[104X
    [4X[28XBasis( <vector space over Rationals, with 6 generators>,[128X[104X
    [4X[28X[ 1*([ 0, 0, 1 ].[ 0, 0, 1 ]), 1*([ 0, 1, 0 ].[ 0, 0, 1 ]), [128X[104X
    [4X[28X  1*([ 0, 1, 0 ].[ 0, 1, 0 ]), 1*([ 1, 0, 0 ].[ 0, 0, 1 ]), [128X[104X
    [4X[28X  1*([ 1, 0, 0 ].[ 0, 1, 0 ]), 1*([ 1, 0, 0 ].[ 1, 0, 0 ])[128X[104X
    [4X[28X ] )[128X[104X
  [4X[32X[104X
  
