61 Vector Spaces
  
  
  61.1 IsLeftVectorSpace (Filter)
  
  61.1-1 IsLeftVectorSpace
  
  IsLeftVectorSpace( V )  Category
  IsVectorSpace( V )  Category
  
  A  vector space in GAP is a free left module (see IsFreeLeftModule (57.3-1))
  over a division ring (see Chapter 58).
  
  Whenever  we  talk  about  an  F-vector  space V then V is an additive group
  (see IsAdditiveGroup  (55.1-6))  on  which  the  division  ring  F  acts via
  multiplication from the left such that this action and the addition in V are
  left and right distributive. The division ring F can be accessed as value of
  the attribute LeftActingDomain (57.1-11).
  
  Vector  spaces  in  GAP are always left vector spaces, IsLeftVectorSpace and
  IsVectorSpace are synonyms.
  
  
  61.2 Constructing Vector Spaces
  
  61.2-1 VectorSpace
  
  VectorSpace( F, gens[, zero][, "basis"] )  function
  
  For  a  field  F  and  a collection gens of vectors, VectorSpace returns the
  F-vector space spanned by the elements in gens.
  
  The  optional  argument  zero can be used to specify the zero element of the
  space;  zero  must  be  given  if gens is empty. The optional string "basis"
  indicates  that  gens  is  known  to  be  linearly  independent  over  F, in
  particular  the  dimension of the vector space is immediately set; note that
  Basis  (61.5-2)  need  not  return  the  basis  formed by gens if the string
  "basis" is given as an argument.
  
    Example  
    gap> V:= VectorSpace( Rationals, [ [ 1, 2, 3 ], [ 1, 1, 1 ] ] );
    <vector space over Rationals, with 2 generators>
  
  
  61.2-2 Subspace
  
  Subspace( V, gens[, "basis"] )  function
  SubspaceNC( V, gens[, "basis"] )  function
  
  For an F-vector space V and a list or collection gens that is a subset of V,
  Subspace  returns  the F-vector space spanned by gens; if gens is empty then
  the  trivial  subspace  (see TrivialSubspace (61.3-2)) of V is returned. The
  parent (see 31.7) of the returned vector space is set to V.
  
  SubspaceNC does the same as Subspace, except that it omits the check whether
  gens is a subset of V.
  
  The  optional  string  "basis"  indicates  that gens is known to be linearly
  independent  over  F.  In  this  case  the  dimension  of  the  subspace  is
  immediately  set, and both Subspace and SubspaceNC do not check whether gens
  really is linearly independent and whether gens is a subset of V.
  
    Example  
    gap> V:= VectorSpace( Rationals, [ [ 1, 2, 3 ], [ 1, 1, 1 ] ] );;
    gap> W:= Subspace( V, [ [ 0, 1, 2 ] ] );
    <vector space over Rationals, with 1 generator>
  
  
  61.2-3 AsVectorSpace
  
  AsVectorSpace( F, D )  operation
  
  Let  F  be  a  division  ring  and  D a domain. If the elements in D form an
  F-vector  space  then  AsVectorSpace  returns this F-vector space, otherwise
  fail is returned.
  
  AsVectorSpace  can  be  used  for  example to view a given vector space as a
  vector space over a smaller or larger division ring.
  
    Example  
    gap> V:= FullRowSpace( GF( 27 ), 3 );
    ( GF(3^3)^3 )
    gap> Dimension( V );  LeftActingDomain( V );
    3
    GF(3^3)
    gap> W:= AsVectorSpace( GF( 3 ), V );
    <vector space over GF(3), with 9 generators>
    gap> Dimension( W );  LeftActingDomain( W );
    9
    GF(3)
    gap> AsVectorSpace( GF( 9 ), V );
    fail
  
  
  61.2-4 AsSubspace
  
  AsSubspace( V, U )  operation
  
  Let  V  be an F-vector space, and U a collection. If U is a subset of V such
  that  the  elements of U form an F-vector space then AsSubspace returns this
  vector  space,  with parent set to V (see AsVectorSpace (61.2-3)). Otherwise
  fail is returned.
  
    Example  
    gap> V:= VectorSpace( Rationals, [ [ 1, 2, 3 ], [ 1, 1, 1 ] ] );;
    gap> W:= VectorSpace( Rationals, [ [ 1/2, 1/2, 1/2 ] ] );;
    gap> U:= AsSubspace( V, W );
    <vector space over Rationals, with 1 generator>
    gap> Parent( U ) = V;
    true
    gap> AsSubspace( V, [ [ 1, 1, 1 ] ] );
    fail
  
  
  
  61.3 Operations and Attributes for Vector Spaces
  
  61.3-1 GeneratorsOfLeftVectorSpace
  
  GeneratorsOfLeftVectorSpace( V )  attribute
  GeneratorsOfVectorSpace( V )  attribute
  
  For  an  F-vector  space  V,  GeneratorsOfLeftVectorSpace  returns a list of
  vectors in V that generate V as an F-vector space.
  
    Example  
    gap> GeneratorsOfVectorSpace( FullRowSpace( Rationals, 3 ) );
    [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ]
  
  
  61.3-2 TrivialSubspace
  
  TrivialSubspace( V )  attribute
  
  For  a  vector  space  V,  TrivialSubspace  returns  the  subspace of V that
  consists of the zero vector in V.
  
    Example  
    gap> V:= GF(3)^3;;
    gap> triv:= TrivialSubspace( V );
    <vector space of dimension 0 over GF(3)>
    gap> AsSet( triv );
    [ [ 0*Z(3), 0*Z(3), 0*Z(3) ] ]
  
  
  
  61.4 Domains of Subspaces of Vector Spaces
  
  61.4-1 Subspaces
  
  Subspaces( V[, k] )  attribute
  
  Called  with  a  finite  vector space v, Subspaces returns the domain of all
  subspaces of V.
  
  Called  with  V and a nonnegative integer k, Subspaces returns the domain of
  all k-dimensional subspaces of V.
  
  Special  Size  (30.4-6) and Iterator (30.8-1) methods are provided for these
  domains.
  
  61.4-2 IsSubspacesVectorSpace
  
  IsSubspacesVectorSpace( D )  Category
  
  The  domain  of all subspaces of a (finite) vector space or of all subspaces
  of  fixed  dimension,  as  returned  by  Subspaces  (61.4-1)  (see Subspaces
  (61.4-1)) lies in the category IsSubspacesVectorSpace.
  
    Example  
    gap> D:= Subspaces( GF(3)^3 );
    Subspaces( ( GF(3)^3 ) )
    gap> Size( D );
    28
    gap> iter:= Iterator( D );;
    gap> NextIterator( iter );
    <vector space of dimension 0 over GF(3)>
    gap> NextIterator( iter );
    <vector space of dimension 1 over GF(3)>
    gap> IsSubspacesVectorSpace( D );
    true
  
  
  
  61.5 Bases of Vector Spaces
  
  In  GAP,  a  basis of a free left F-module V is a list of vectors B = [ v_1,
  v_2,  ...,  v_n  ] in V such that V is generated as a left F-module by these
  vectors and such that B is linearly independent over F. The integer n is the
  dimension  of  V (see Dimension (57.3-3)). In particular, as each basis is a
  list  (see Chapter 21), it has a length (see Length (21.17-5)), and the i-th
  vector of B can be accessed as B[i].
  
    Example  
    gap> V:= Rationals^3;
    ( Rationals^3 )
    gap> B:= Basis( V );
    CanonicalBasis( ( Rationals^3 ) )
    gap> Length( B );
    3
    gap> B[1];
    [ 1, 0, 0 ]
  
  
  The  operations  described  below  make  sense  only  for  bases  of  finite
  dimensional  vector  spaces.  (In practice this means that the vector spaces
  must  be  low  dimensional,  that  is, the dimension should not exceed a few
  hundred.)
  
  Besides  the basic operations for lists (see 21.2), the basic operations for
  bases  are  BasisVectors  (61.6-1), Coefficients (61.6-3), LinearCombination
  (61.6-4),  and UnderlyingLeftModule (61.6-2). These and other operations for
  arbitrary bases are described in 61.6.
  
  For special kinds of bases, further operations are defined (see 61.7).
  
  GAP supports the following three kinds of bases.
  
  Relative  bases  delegate  the  work  to  other  bases of the same free left
  module, via basechange matrices (see RelativeBasis (61.5-4)).
  
  Bases  handled  by  nice bases delegate the work to bases of isomorphic left
  modules over the same left acting domain (see 61.11).
  
  Finally, of course there must be bases in GAP that really do the work.
  
  For  example,  in  the  case of a Gaussian row or matrix space V (see 61.9),
  Basis( V ) is a semi-echelonized basis (see IsSemiEchelonized (61.9-7)) that
  uses  Gaussian  elimination;  such  a  basis is of the third kind. Basis( V,
  vectors  ) 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.
  
  Bases   handled   by   nice  bases  are  described  in 61.11.  Examples  are
  non-Gaussian  row  and  matrix  spaces,  and  subspaces of finite fields and
  abelian number fields that are themselves not fields.
  
  61.5-1 IsBasis
  
  IsBasis( obj )  Category
  
  In GAP, a basis of a free left module is an object that knows how to compute
  coefficients w.r.t. its basis vectors (see Coefficients (61.6-3)). Bases are
  constructed  by  Basis  (61.5-2).  Each basis is an immutable list, the i-th
  entry being the i-th basis vector.
  
  (See 61.8 for mutable bases.)
  
    Example  
    gap> V:= GF(2)^2;;
    gap> B:= Basis( V );;
    gap> IsBasis( B );
    true
    gap> IsBasis( [ [ 1, 0 ], [ 0, 1 ] ] );
    false
    gap> IsBasis( Basis( Rationals^2, [ [ 1, 0 ], [ 0, 1 ] ] ) );
    true
  
  
  61.5-2 Basis
  
  Basis( V[, vectors] )  attribute
  BasisNC( V, vectors )  operation
  
  Called  with  a  free left F-module V as the only argument, Basis returns an
  F-basis of V whose vectors are not further specified.
  
  If  additionally  a  list  vectors  of  vectors  in V is given that forms an
  F-basis  of  V  then  Basis  returns  this basis; if vectors is not linearly
  independent  over F or does not generate V as a free left F-module then fail
  is returned.
  
  BasisNC  does  the same as the two argument version of Basis, except that it
  does not check whether vectors form a basis.
  
  If  no  basis  vectors  are  prescribed  then  Basis  need not compute basis
  vectors;  in  this  case,  the  vectors  are  computed  in the first call to
  BasisVectors (61.6-1).
  
    Example  
    gap> V:= VectorSpace( Rationals, [ [ 1, 2, 7 ], [ 1/2, 1/3, 5 ] ] );;
    gap> B:= Basis( V );
    SemiEchelonBasis( <vector space over Rationals, with 
    2 generators>, ... )
    gap> BasisVectors( B );
    [ [ 1, 2, 7 ], [ 0, 1, -9/4 ] ]
    gap> B:= Basis( V, [ [ 1, 2, 7 ], [ 3, 2, 30 ] ] );
    Basis( <vector space over Rationals, with 2 generators>, 
    [ [ 1, 2, 7 ], [ 3, 2, 30 ] ] )
    gap> Basis( V, [ [ 1, 2, 3 ] ] );
    fail
  
  
  61.5-3 CanonicalBasis
  
  CanonicalBasis( V )  attribute
  
  If the vector space V supports a canonical basis then CanonicalBasis returns
  this basis, otherwise fail is returned.
  
  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 LeftActingDomain (57.1-11))
  then  the  equality  of  these vector spaces can be decided by comparing the
  canonical bases.
  
  The  exact  meaning of a canonical basis depends on the type of V. Canonical
  bases are defined for example for Gaussian row and matrix spaces (see 61.9).
  
  If  one  designs  a  new  kind  of  vector  spaces (see 61.12) and defines a
  canonical basis for these spaces then the CanonicalBasis method one installs
  (see InstallMethod  (78.3-1))  must  not  call  Basis (61.5-2). On the other
  hand,  one probably should install a Basis (61.5-2) method that simply calls
  CanonicalBasis,   the   value  of  the  method  (see 78.3  and  78.4)  being
  CANONICAL_BASIS_FLAGS.
  
    Example  
    gap> vecs:= [ [ 1, 2, 3 ], [ 1, 1, 1 ], [ 1, 1, 1 ] ];;
    gap> V:= VectorSpace( Rationals, vecs );;
    gap> B:= CanonicalBasis( V );
    CanonicalBasis( <vector space over Rationals, with 3 generators> )
    gap> BasisVectors( B );
    [ [ 1, 0, -1 ], [ 0, 1, 2 ] ]
  
  
  61.5-4 RelativeBasis
  
  RelativeBasis( B, vectors )  operation
  RelativeBasisNC( B, vectors )  operation
  
  A  relative  basis  is  a basis of the free left module V that delegates the
  computation  of  coefficients  etc.  to  another basis of V via a basechange
  matrix.
  
  Let B be a basis of the free left module V, and vectors a list of vectors in
  V.
  
  RelativeBasis  checks  whether vectors form a basis of V, and in this case a
  basis  is returned in which vectors are the basis vectors; otherwise fail is
  returned.
  
  RelativeBasisNC does the same, except that it omits the check.
  
  
  61.6 Operations for Vector Space Bases
  
  61.6-1 BasisVectors
  
  BasisVectors( B )  attribute
  
  For  a  vector space basis B, BasisVectors returns the list of basis vectors
  of  B.  The  lists  B  and  BasisVectors( B ) are equal; the main purpose of
  BasisVectors  is  to  provide access to a list of vectors that does not know
  about an underlying vector space.
  
    Example  
    gap> V:= VectorSpace( Rationals, [ [ 1, 2, 7 ], [ 1/2, 1/3, 5 ] ] );;
    gap> B:= Basis( V, [ [ 1, 2, 7 ], [ 0, 1, -9/4 ] ] );;
    gap> BasisVectors( B );
    [ [ 1, 2, 7 ], [ 0, 1, -9/4 ] ]
  
  
  61.6-2 UnderlyingLeftModule
  
  UnderlyingLeftModule( B )  attribute
  
  For a basis B of a free left module V, UnderlyingLeftModule returns V.
  
  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 Basis (61.5-2)); furthermore, in
  some  cases  it  is  convenient to test membership of a vector in the module
  before computing coefficients w.r.t. a basis.
  
    Example  
    gap> B:= Basis( GF(2)^6 );;  UnderlyingLeftModule( B );
    ( GF(2)^6 )
  
  
  61.6-3 Coefficients
  
  Coefficients( B, v )  operation
  
  Let V be the underlying left module of the basis B, and v a vector such that
  the  family  of  v  is  the  elements  family  of  the  family  of  V.  Then
  Coefficients(  B, v ) is the list of coefficients of v w.r.t. B if v lies in
  V, and fail otherwise.
  
    Example  
    gap> V:= VectorSpace( Rationals, [ [ 1, 2, 7 ], [ 1/2, 1/3, 5 ] ] );;
    gap> B:= Basis( V, [ [ 1, 2, 7 ], [ 0, 1, -9/4 ] ] );;
    gap> Coefficients( B, [ 1/2, 1/3, 5 ] );
    [ 1/2, -2/3 ]
    gap> Coefficients( B, [ 1, 0, 0 ] );
    fail
  
  
  61.6-4 LinearCombination
  
  LinearCombination( B, coeff )  operation
  
  If  B  is  a  basis  object  (see IsBasis (61.5-1)) or a homogeneous list of
  length  n,  and  coeff is a row vector of the same length, LinearCombination
  returns the vector ∑_{i = 1}^n coeff[i] * B[i].
  
  Perhaps the most important usage is the case where B forms a basis.
  
    Example  
    gap> V:= VectorSpace( Rationals, [ [ 1, 2, 7 ], [ 1/2, 1/3, 5 ] ] );;
    gap> B:= Basis( V, [ [ 1, 2, 7 ], [ 0, 1, -9/4 ] ] );;
    gap> LinearCombination( B, [ 1/2, -2/3 ] );
    [ 1/2, 1/3, 5 ]
  
  
  61.6-5 EnumeratorByBasis
  
  EnumeratorByBasis( B )  attribute
  
  For  a basis B of the free left F-module V of dimension n, EnumeratorByBasis
  returns  an  enumerator  that  loops  over  the  elements  of  V  as  linear
  combinations  of  the  vectors of B with coefficients the row vectors in the
  full  row  space  (see FullRowSpace  (61.9-4)) of dimension n over F, in the
  succession given by the default enumerator of this row space.
  
    Example  
    gap> V:= GF(2)^3;;
    gap> enum:= EnumeratorByBasis( CanonicalBasis( V ) );;
    gap> Print( enum{ [ 1 .. 4 ] }, "\n" );
    [ [ 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 ] ]
    gap> B:= Basis( V, [ [ 1, 1, 1 ], [ 1, 1, 0 ], [ 1, 0, 0 ] ] * Z(2) );;
    gap> enum:= EnumeratorByBasis( B );;
    gap> Print( enum{ [ 1 .. 4 ] }, "\n" );
    [ [ 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) ] ]
  
  
  61.6-6 IteratorByBasis
  
  IteratorByBasis( B )  operation
  
  For  a  basis  B of the free left F-module V of dimension n, IteratorByBasis
  returns an iterator that loops over the elements of V as linear combinations
  of  the vectors of B with coefficients the row vectors in the full row space
  (see FullRowSpace  (61.9-4))  of dimension n over F, in the succession given
  by the default enumerator of this row space.
  
    Example  
    gap> V:= GF(2)^3;;
    gap> iter:= IteratorByBasis( CanonicalBasis( V ) );;
    gap> for i in [ 1 .. 4 ] do Print( NextIterator( iter ), "\n" ); od;
    [ 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 ]
    gap> B:= Basis( V, [ [ 1, 1, 1 ], [ 1, 1, 0 ], [ 1, 0, 0 ] ] * Z(2) );;
    gap> iter:= IteratorByBasis( B );;
    gap> for i in [ 1 .. 4 ] do Print( NextIterator( iter ), "\n" ); od;
    [ 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) ]
  
  
  
  61.7 Operations for Special Kinds of Bases
  
  61.7-1 IsCanonicalBasis
  
  IsCanonicalBasis( B )  property
  
  If  the  underlying  free  left module V of the basis B supports a canonical
  basis  (see CanonicalBasis (61.5-3)) then IsCanonicalBasis returns true if B
  is equal to the canonical basis of V, and false otherwise.
  
  61.7-2 IsIntegralBasis
  
  IsIntegralBasis( B )  property
  
  Let  B  be an S-basis of a field F for a subfield S of F, and let R and M be
  the  rings  of  algebraic integers in S and F, respectively. IsIntegralBasis
  returns true if B is also an R-basis of M, and false otherwise.
  
  61.7-3 IsNormalBasis
  
  IsNormalBasis( B )  property
  
  Let  B  be  an  S-basis  of  a  field F for a subfield S of F. IsNormalBasis
  returns  true  if  B  is  invariant  under the Galois group (see GaloisGroup
  (58.3-1)) of the field extension F / S, and false otherwise.
  
    Example  
    gap> B:= CanonicalBasis( GaussianRationals );
    CanonicalBasis( GaussianRationals )
    gap> IsIntegralBasis( B );  IsNormalBasis( B );
    true
    false
  
  
  
  61.8 Mutable Bases
  
  It  is  useful  to  have  a mutable basis 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.
  
  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 UnderlyingLeftModule (61.6-2)).
  
  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.
  
  Since  Immutable  produces  an immutable copy of any GAP 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  mutable GAP objects, hence these objects are unable to store attribute
  values.
  
  Basic   operations   for   immutable   bases  are  NrBasisVectors  (61.8-3),
  IsContainedInSpan   (61.8-5),   CloseMutableBasis  (61.8-6),  ImmutableBasis
  (61.8-4),  Coefficients  (61.6-3),  and  BasisVectors  (61.6-1). ShallowCopy
  (12.7-1)  for  a  mutable  basis returns a mutable plain list containing the
  current basis vectors.
  
  Since  mutable  bases do not admit arbitrary changes of their lists of basis
  vectors,  a  mutable basis is not a list. It is, however, a collection, more
  precisely its family (see 13.1) equals the family of its collection of basis
  vectors.
  
  Mutable bases can be constructed with MutableBasis.
  
  Similar  to  the situation with bases (cf. 61.5), GAP supports the following
  three kinds of mutable bases.
  
  The  generic  method  of  MutableBasis  returns  a mutable basis that simply
  stores an immutable basis; clearly one wants to avoid this whenever possible
  with reasonable effort.
  
  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 61.11)  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.
  
  There are mutable bases that use special information to perform their tasks;
  examples are mutable bases of Gaussian row and matrix spaces.
  
  61.8-1 IsMutableBasis
  
  IsMutableBasis( MB )  Category
  
  Every mutable basis lies in the category IsMutableBasis.
  
  61.8-2 MutableBasis
  
  MutableBasis( R, vectors[, zero] )  operation
  
  MutableBasis  returns a mutable basis for the R-free module generated by the
  vectors  in  the list vectors. The optional argument zero is the zero vector
  of the module; it must be given if vectors is empty.
  
  Note  that  vectors  will in general not be the basis vectors of the mutable
  basis!
  
    Example  
    gap> MB:= MutableBasis( Rationals, [ [ 1, 2, 3 ], [ 0, 1, 0 ] ] );
    <mutable basis over Rationals, 2 vectors>
  
  
  61.8-3 NrBasisVectors
  
  NrBasisVectors( MB )  operation
  
  For  a  mutable basis MB, NrBasisVectors returns the current number of basis
  vectors  of MB. Note that this operation is not an attribute, as it makes no
  sense  to store the value. NrBasisVectors is used mainly as an equivalent of
  Dimension for the underlying left module in the case of immutable bases.
  
    Example  
    gap> MB:= MutableBasis( Rationals, [ [ 1, 1], [ 2, 2 ] ] );;
    gap> NrBasisVectors( MB );
    1
  
  
  61.8-4 ImmutableBasis
  
  ImmutableBasis( MB[, V] )  operation
  
  ImmutableBasis  returns the immutable basis B with the same basis vectors as
  in the mutable basis MB.
  
  If   the   second   argument   V   is   present  then  V  is  the  value  of
  UnderlyingLeftModule  (61.6-2)  for B. 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 MB, for example if it is
  an algebra.
  
    Example  
    gap> MB:= MutableBasis( Rationals, [ [ 1, 1 ], [ 2, 2 ] ] );;
    gap> B:= ImmutableBasis( MB );
    SemiEchelonBasis( <vector space of dimension 1 over Rationals>, 
    [ [ 1, 1 ] ] )
    gap> UnderlyingLeftModule( B );
    <vector space of dimension 1 over Rationals>
  
  
  61.8-5 IsContainedInSpan
  
  IsContainedInSpan( MB, v )  operation
  
  For  a  mutable  basis  MB  over  the  coefficient  ring  R  and a vector v,
  IsContainedInSpan  returns true is v lies in the R-span of the current basis
  vectors of MB, and false otherwise.
  
  61.8-6 CloseMutableBasis
  
  CloseMutableBasis( MB, v )  operation
  
  For  a  mutable  basis  MB  over  the  coefficient  ring  R  and a vector v,
  CloseMutableBasis changes MB such that afterwards it describes the R-span of
  the former basis vectors together with v.
  
  Note  that  if  v  enlarges the dimension then this does in general not mean
  that  v  is  simply  added  to  the  basis  vectors  of MB. Usually a linear
  combination  of  v  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 IsSemiEchelonized (61.9-7)).
  
  CloseMutableBasis  returns false if v was already in the R-span described by
  MB, and true if MB got extended.
  
    Example  
    gap> MB:= MutableBasis( Rationals, [ [ 1, 1, 3 ], [ 2, 2, 1 ] ] );
    <mutable basis over Rationals, 2 vectors>
    gap> IsContainedInSpan( MB, [ 1, 0, 0 ] );
    false
    gap> CloseMutableBasis( MB, [ 1, 0, 0 ] );
    true
    gap> MB;
    <mutable basis over Rationals, 3 vectors>
    gap> IsContainedInSpan( MB, [ 1, 0, 0 ] );
    true
    gap> CloseMutableBasis( MB, [ 1, 0, 0 ] );
    false
  
  
  
  61.9 Row and Matrix Spaces
  
  61.9-1 IsRowSpace
  
  IsRowSpace( V )  filter
  
  A  row  space  in  GAP  is  a vector space that consists of row vectors (see
  Chapter 23).
  
  61.9-2 IsMatrixSpace
  
  IsMatrixSpace( V )  filter
  
  A  matrix  space  in  GAP  is  a vector space that consists of matrices (see
  Chapter 24).
  
  61.9-3 IsGaussianSpace
  
  IsGaussianSpace( V )  filter
  
  The  filter  IsGaussianSpace  (see 13.2)  for  the row space (see IsRowSpace
  (61.9-1))  or  matrix  space  (see IsMatrixSpace  (61.9-2)) V over a field F
  indicates   that   the  entries  of  all  row  vectors  or  matrices  in  V,
  respectively,  are  all contained in F. In this case, V is called a Gaussian
  vector  space.  Bases  for  Gaussian  spaces  can be computed using Gaussian
  elimination for a given list of vector space generators.
  
    Example  
    gap> mats:= [ [[1,1],[2,2]], [[3,4],[0,1]] ];;
    gap> V:= VectorSpace( Rationals, mats );;
    gap> IsGaussianSpace( V );
    true
    gap> mats[1][1][1]:= E(4);;   # an element in an extension field
    gap> V:= VectorSpace( Rationals, mats );;
    gap> IsGaussianSpace( V );
    false
    gap> V:= VectorSpace( Field( Rationals, [ E(4) ] ), mats );;
    gap> IsGaussianSpace( V );
    true
  
  
  61.9-4 FullRowSpace
  
  FullRowSpace( F, n )  function
  \^( F, n )  method
  
  For a field F and a nonnegative integer n, FullRowSpace returns the F-vector
  space  that consists of all row vectors (see IsRowVector (23.1-1)) of length
  n with entries in F.
  
  An alternative to construct this vector space is via F^n.
  
    Example  
    gap> FullRowSpace( GF( 9 ), 3 );
    ( GF(3^2)^3 )
    gap> GF(9)^3;           # the same as above
    ( GF(3^2)^3 )
  
  
  61.9-5 FullMatrixSpace
  
  FullMatrixSpace( F, m, n )  function
  \^( F, dims )  method
  
  For a field F and two positive integers m and n, FullMatrixSpace returns the
  F-vector  space that consists of all m by n matrices (see IsMatrix (24.2-1))
  with entries in F.
  
  If  m  =  n  then  the  result  is in fact an algebra (see FullMatrixAlgebra
  (62.5-4)).
  
  An alternative to construct this vector space is via F^[m,n].
  
    Example  
    gap> FullMatrixSpace( GF(2), 4, 5 );
    ( GF(2)^[ 4, 5 ] )
    gap> GF(2)^[ 4, 5 ];    # the same as above
    ( GF(2)^[ 4, 5 ] )
  
  
  61.9-6 DimensionOfVectors
  
  DimensionOfVectors( M )  attribute
  
  For a left module M that consists of row vectors (see IsRowModule (57.3-6)),
  DimensionOfVectors  returns the common length of all row vectors in M. For a
  left  module  M  that  consists  of  matrices (see IsMatrixModule (57.3-7)),
  DimensionOfVectors  returns  the common matrix dimensions (see DimensionsMat
  (24.4-1)) of all matrices in M.
  
    Example  
    gap> DimensionOfVectors( GF(2)^5 );
    5
    gap> DimensionOfVectors( GF(2)^[2,3] );
    [ 2, 3 ]
  
  
  61.9-7 IsSemiEchelonized
  
  IsSemiEchelonized( B )  property
  
  Let  B  be  a basis of a Gaussian row or matrix space V (see IsGaussianSpace
  (61.9-3)) over the field F.
  
  If  V  is a row space then B 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 F, and all values exactly below these pivot elements are the
  zero of F (cf. SemiEchelonMat (24.10-1)).
  
  If  V is a matrix space then B is semi-echelonized if the matrix obtained by
  replacing   each   basis   vector  by  the  concatenation  of  its  rows  is
  semi-echelonized (see above, cf. SemiEchelonMats (24.10-4)).
  
    Example  
    gap> V:= GF(2)^2;;
    gap> B1:= Basis( V, [ [ 0, 1 ], [ 1, 0 ] ] * Z(2) );;
    gap> IsSemiEchelonized( B1 );
    true
    gap> B2:= Basis( V, [ [ 0, 1 ], [ 1, 1 ] ] * Z(2) );;
    gap> IsSemiEchelonized( B2 );
    false
  
  
  61.9-8 SemiEchelonBasis
  
  SemiEchelonBasis( V[, vectors] )  attribute
  SemiEchelonBasisNC( V, vectors )  operation
  
  Let  V  be  a  Gaussian  row  or  matrix  vector  space  over  the  field  F
  (see IsGaussianSpace (61.9-3), IsRowSpace (61.9-1), IsMatrixSpace (61.9-2)).
  
  Called  with  V  as the only argument, SemiEchelonBasis returns a basis of V
  that has the property IsSemiEchelonized (61.9-7).
  
  If  additionally  a  list  vectors  of  vectors  in  V is given that forms a
  semi-echelonized  basis  of  V  then SemiEchelonBasis returns this basis; if
  vectors do not form a basis of V then fail is returned.
  
  SemiEchelonBasisNC   does   the   same   as  the  two  argument  version  of
  SemiEchelonBasis,  except  that  it  is  not  checked whether vectors form a
  semi-echelonized basis.
  
    Example  
    gap> V:= GF(2)^2;;
    gap> B:= SemiEchelonBasis( V );
    SemiEchelonBasis( ( GF(2)^2 ), ... )
    gap> Print( BasisVectors( B ), "\n" );
    [ [ Z(2)^0, 0*Z(2) ], [ 0*Z(2), Z(2)^0 ] ]
    gap> B:= SemiEchelonBasis( V, [ [ 1, 1 ], [ 0, 1 ] ] * Z(2) );
    SemiEchelonBasis( ( GF(2)^2 ), <an immutable 2x2 matrix over GF2> )
    gap> Print( BasisVectors( B ), "\n" );
    [ [ Z(2)^0, Z(2)^0 ], [ 0*Z(2), Z(2)^0 ] ]
    gap> Coefficients( B, [ 0, 1 ] * Z(2) );
    [ 0*Z(2), Z(2)^0 ]
    gap> Coefficients( B, [ 1, 0 ] * Z(2) );
    [ Z(2)^0, Z(2)^0 ]
    gap> SemiEchelonBasis( V, [ [ 0, 1 ], [ 1, 1 ] ] * Z(2) );
    fail
  
  
  61.9-9 IsCanonicalBasisFullRowModule
  
  IsCanonicalBasisFullRowModule( B )  property
  
  IsCanonicalBasisFullRowModule  returns  true  if  B  is  the canonical basis
  (see IsCanonicalBasis  (61.7-1))  of  a full row module (see IsFullRowModule
  (57.3-8)), and false otherwise.
  
  The  canonical  basis  of  a  Gaussian  row  space  is defined as the unique
  semi-echelonized  (see IsSemiEchelonized (61.9-7)) basis with the additional
  property  that  for  j > i the position of the pivot of row j is bigger than
  the  position  of  the  pivot  of row i, and that each pivot column contains
  exactly one nonzero entry.
  
  61.9-10 IsCanonicalBasisFullMatrixModule
  
  IsCanonicalBasisFullMatrixModule( B )  property
  
  IsCanonicalBasisFullMatrixModule  returns  true  if B is the canonical basis
  (see IsCanonicalBasis     (61.7-1))     of     a    full    matrix    module
  (see IsFullMatrixModule (57.3-10)), and false otherwise.
  
  The  canonical  basis  of  a  Gaussian matrix space is defined as the unique
  semi-echelonized  (see IsSemiEchelonized  (61.9-7)) basis for which the list
  of  concatenations  of  the  basis  vectors forms the canonical basis of the
  corresponding Gaussian row space.
  
  61.9-11 NormedRowVectors
  
  NormedRowVectors( V )  attribute
  
  For  a finite Gaussian row space V (see IsRowSpace (61.9-1), IsGaussianSpace
  (61.9-3)),  NormedRowVectors  returns  a  list of those nonzero vectors in V
  that have a one in the first nonzero component.
  
  The  result  list  can  be  used as action domain for the action of a matrix
  group   via   OnLines   (41.2-12),   which  yields  the  natural  action  on
  one-dimensional subspaces of V (see also Subspaces (61.4-1)).
  
    Example  
    gap> vecs:= NormedRowVectors( GF(3)^2 );
    [ [ 0*Z(3), Z(3)^0 ], [ Z(3)^0, 0*Z(3) ], [ Z(3)^0, Z(3)^0 ], 
      [ Z(3)^0, Z(3) ] ]
    gap> Action( GL(2,3), vecs, OnLines );
    Group([ (3,4), (1,2,4) ])
  
  
  61.9-12 SiftedVector
  
  SiftedVector( B, v )  operation
  
  Let  B  be  a  semi-echelonized  basis (see IsSemiEchelonized (61.9-7)) of a
  Gaussian  row  or matrix space V (see IsGaussianSpace (61.9-3)), and v a row
  vector  or matrix, respectively, of the same dimension as the elements in V.
  SiftedVector  returns the residuum of v with respect to B, which is obtained
  by  successively  cleaning the pivot positions in v by subtracting multiples
  of the basis vectors in B. So the result is the zero vector in V if and only
  if v lies in V.
  
  B may also be a mutable basis (see 61.8) of a Gaussian row or matrix space.
  
    Example  
    gap> V:= VectorSpace( Rationals, [ [ 1, 2, 7 ], [ 1/2, 1/3, 5 ] ] );;
    gap> B:= Basis( V );;
    gap> SiftedVector( B, [ 1, 2, 8 ] );
    [ 0, 0, 1 ]
  
  
  
  61.10 Vector Space Homomorphisms
  
  Vector   space   homomorphisms   (or   linear   mappings)   are  defined  in
  Section 32.11.  GAP  provides  special  functions  to construct a particular
  linear mapping from images of given elements in the source, from a matrix of
  coefficients, or as a natural epimorphism.
  
  F-linear  mappings  with same source and same range can be added, so one can
  form vector spaces of linear mappings.
  
  61.10-1 LeftModuleGeneralMappingByImages
  
  LeftModuleGeneralMappingByImages( V, W, gens, imgs )  operation
  
  Let  V and W be two left modules over the same left acting domain R and gens
  and  imgs  lists  (of the same length) of elements in V and W, respectively.
  LeftModuleGeneralMappingByImages  returns  the general mapping with source V
  and  range  W  that  is  defined  by  mapping  the  elements  in gens to the
  corresponding elements in imgs, and taking the R-linear closure.
  
  gens  need  not generate V as a left R-module, and if the specification does
  not  define  a linear mapping then the result will be multi-valued; hence in
  general it is not a mapping (see IsMapping (32.3-3)).
  
    Example  
    gap> V:= Rationals^2;;
    gap> W:= VectorSpace( Rationals, [ [1,2,3], [1,0,1] ] );;
    gap> f:= LeftModuleGeneralMappingByImages( V, W,
    >                                [[1,0],[2,0]], [[1,0,1],[1,0,1] ] );
    [ [ 1, 0 ], [ 2, 0 ] ] -> [ [ 1, 0, 1 ], [ 1, 0, 1 ] ]
    gap> IsMapping( f );
    false
  
  
  61.10-2 LeftModuleHomomorphismByImages
  
  LeftModuleHomomorphismByImages( V, W, gens, imgs )  function
  LeftModuleHomomorphismByImagesNC( V, W, gens, imgs )  operation
  
  Let  V and W be two left modules over the same left acting domain R and gens
  and  imgs  lists  (of the same length) of elements in V and W, respectively.
  LeftModuleHomomorphismByImages  returns  the left R-module homomorphism with
  source  V and range W that is defined by mapping the elements in gens to the
  corresponding elements in imgs.
  
  If  gens does not generate V or if the homomorphism does not exist (i.e., if
  mapping  the  generators describes only a multi-valued mapping) then fail is
  returned.  For  creating  a  possibly  multi-valued mapping from V to W that
  respects    addition,    multiplication,    and    scalar    multiplication,
  LeftModuleGeneralMappingByImages (61.10-1) can be used.
  
  LeftModuleHomomorphismByImagesNC        does        the        same       as
  LeftModuleHomomorphismByImages, except that it omits all checks.
  
    Example  
    gap> V:=Rationals^2;;
    gap> W:=VectorSpace( Rationals, [ [ 1, 0, 1 ], [ 1, 2, 3 ] ] );;
    gap> f:=LeftModuleHomomorphismByImages( V, W,
    > [ [ 1, 0 ], [ 0, 1 ] ], [ [ 1, 0, 1 ], [ 1, 2, 3 ] ] );
    [ [ 1, 0 ], [ 0, 1 ] ] -> [ [ 1, 0, 1 ], [ 1, 2, 3 ] ]
    gap> Image( f, [1,1] );
    [ 2, 2, 4 ]
  
  
  61.10-3 LeftModuleHomomorphismByMatrix
  
  LeftModuleHomomorphismByMatrix( BS, matrix, BR )  operation
  
  Let  BS  and  BR  be  bases  of  the  left  R-modules V and W, respectively.
  LeftModuleHomomorphismByMatrix returns the R-linear mapping from V to W that
  is  defined  by  the  matrix matrix, as follows. The image of the i-th basis
  vector  of  BS  is  the  linear  combination of the basis vectors of BR with
  coefficients the i-th row of matrix.
  
    Example  
    gap> V:= Rationals^2;;
    gap> W:= VectorSpace( Rationals, [ [ 1, 0, 1 ], [ 1, 2, 3 ] ] );;
    gap> f:= LeftModuleHomomorphismByMatrix( Basis( V ),
    > [ [ 1, 2 ], [ 3, 1 ] ], Basis( W ) );
    <linear mapping by matrix, ( Rationals^
    2 ) -> <vector space over Rationals, with 2 generators>>
  
  
  61.10-4 NaturalHomomorphismBySubspace
  
  NaturalHomomorphismBySubspace( V, W )  operation
  
  For an R-vector space V and a subspace W of V, NaturalHomomorphismBySubspace
  returns  the  R-linear  mapping that is the natural projection of V onto the
  factor space V / W.
  
    Example  
    gap> V:= Rationals^3;;
    gap> W:= VectorSpace( Rationals, [ [ 1, 1, 1 ] ] );;
    gap> f:= NaturalHomomorphismBySubspace( V, W );
    <linear mapping by matrix, ( Rationals^3 ) -> ( Rationals^2 )>
  
  
  61.10-5 Hom
  
  Hom( F, V, W )  operation
  
  For  a  field  F  and  two  vector  spaces  V  and W that can be regarded as
  F-modules (see AsLeftModule (57.1-5)), Hom returns the F-vector space of all
  F-linear mappings from V to W.
  
    Example  
    gap> V:= Rationals^2;;
    gap> W:= VectorSpace( Rationals, [ [ 1, 0, 1 ], [ 1, 2, 3 ] ] );;
    gap> H:= Hom( Rationals, V, W );
    Hom( Rationals, ( Rationals^2 ), <vector space over Rationals, with 
    2 generators> )
    gap> Dimension( H );
    4
  
  
  61.10-6 End
  
  End( F, V )  operation
  
  For  a  field  F  and  a  vector space V that can be regarded as an F-module
  (see AsLeftModule  (57.1-5)),  End  returns  the  F-algebra  of all F-linear
  mappings from V to V.
  
    Example  
    gap> A:= End( Rationals, Rationals^2 );
    End( Rationals, ( Rationals^2 ) )
    gap> Dimension( A );
    4
  
  
  61.10-7 IsFullHomModule
  
  IsFullHomModule( M )  property
  
  A  full  hom  module  is  a module of all R-linear mappings between two left
  R-modules.  The  function  Hom (61.10-5) can be used to construct a full hom
  module.
  
    Example  
    gap> V:= Rationals^2;;
    gap> W:= VectorSpace( Rationals, [ [ 1, 0, 1 ], [ 1, 2, 3 ] ] );;
    gap> H:= Hom( Rationals, V, W );;
    gap> IsFullHomModule( H );
    true
  
  
  61.10-8 IsPseudoCanonicalBasisFullHomModule
  
  IsPseudoCanonicalBasisFullHomModule( B )  property
  
  A  basis  of  a  full  hom  module  is  called pseudo canonical basis if the
  matrices  of  its  basis vectors w.r.t. the stored bases of source and range
  contain exactly one identity entry and otherwise zeros.
  
  Note  that  this  is  not  a  canonical  basis (see CanonicalBasis (61.5-3))
  because it depends on the stored bases of source and range.
  
    Example  
    gap> IsPseudoCanonicalBasisFullHomModule( Basis( H ) );
    true
  
  
  61.10-9 IsLinearMappingsModule
  
  IsLinearMappingsModule( V )  filter
  
  If  an  F-vector  space  V is in the filter IsLinearMappingsModule then this
  expresses  that V consists of linear mappings, and that V is handled via the
  mechanism  of  nice  bases (see 61.11), in the following way. Let S and R be
  the  source  and  the  range,  respectively,  of each mapping in V. Then the
  NiceFreeLeftModuleInfo  (61.11-3) value of V is a record with the components
  basissource  (a  basis  B_S of S) and basisrange (a basis B_R of R), and the
  NiceVector (61.11-2) value of v ∈ V is defined as the matrix of the F-linear
  mapping v w.r.t. the bases B_S and B_R.
  
  
  61.11 Vector Spaces Handled By Nice Bases
  
  There  are  kinds  of  free  R-modules  for which efficient computations are
  possible  because  the  elements are nice, for example subspaces of full row
  modules or of full matrix modules. In other cases, a nice canonical basis is
  known  that allows one to do the necessary computations in the corresponding
  row module, for example algebras given by structure constants.
  
  In  many  other situations, one knows at least an isomorphism from the given
  module V to a nicer free left module W, in the sense that for each vector in
  V, the image in W can easily be computed, and analogously for each vector in
  W, one can compute the preimage in V.
  
  This  allows  one  to  delegate  computations  w.r.t. a  basis B of V to the
  corresponding  basis C of W. We call W the nice free left module of V, and C
  the  nice  basis  of  B.  (Note  that  it  may  happen that also C delegates
  questions to a nicer basis.) The basis B indicates the intended behaviour by
  the  filter  IsBasisByNiceBasis  (61.11-5),  and  stores  C  as value of the
  attribute  NiceBasis  (61.11-4).  V  indicates the intended behaviour by the
  filter  IsHandledByNiceBasis  (61.11-6),  and  stores  W  as  value  of  the
  attribute NiceFreeLeftModule (61.11-1).
  
  The  bijection  between  V  and W is implemented by the functions NiceVector
  (61.11-2) and UglyVector (61.11-2); additional data needed to compute images
  and preimages can be stored as value of NiceFreeLeftModuleInfo (61.11-3).
  
  61.11-1 NiceFreeLeftModule
  
  NiceFreeLeftModule( V )  attribute
  
  For  a  free  left module V that is handled via the mechanism of nice bases,
  this attribute stores the associated free left module to which the tasks are
  delegated.
  
  61.11-2 NiceVector
  
  NiceVector( V, v )  operation
  UglyVector( V, r )  operation
  
  NiceVector and UglyVector provide the linear bijection between the free left
  module V and W:= NiceFreeLeftModule( V ).
  
  If  v lies in the elements family of the family of V then NiceVector( v ) is
  either fail or an element in the elements family of the family of W.
  
  If  r lies in the elements family of the family of W then UglyVector( r ) is
  either fail or an element in the elements family of the family of V.
  
  If  v  lies  in  V  (which  usually  cannot be checked without using W) then
  UglyVector(  V,  NiceVector( V, v ) ) = v. If r lies in W (which usually can
  be checked) then NiceVector( V, UglyVector( V, r ) ) = r.
  
  (This  allows one to implement for example a membership test for V using the
  membership test in W.)
  
  61.11-3 NiceFreeLeftModuleInfo
  
  NiceFreeLeftModuleInfo( V )  attribute
  
  For  a  free  left module V that is handled via the mechanism of nice bases,
  this  operation  has to provide the necessary information (if any) for calls
  of NiceVector (61.11-2) and UglyVector (61.11-2).
  
  61.11-4 NiceBasis
  
  NiceBasis( B )  attribute
  
  Let  B be a basis of a free left module V that is handled via nice bases. If
  B  has  no  basis  vectors stored at the time of the first call to NiceBasis
  then  NiceBasis(  B  )  is  obtained as Basis( NiceFreeLeftModule( V ) ). If
  basis  vectors  are  stored then NiceBasis( B ) is the result of the call of
  Basis  with  arguments  NiceFreeLeftModule( V ) and the NiceVector values of
  the basis vectors of B.
  
  Note  that  the  result is fail if and only if the basis vectors stored in B
  are in fact not basis vectors.
  
  The  attributes  GeneratorsOfLeftModule  of the underlying left modules of B
  and  the  result  of  NiceBasis  correspond  via  NiceVector  (61.11-2)  and
  UglyVector (61.11-2).
  
  61.11-5 IsBasisByNiceBasis
  
  IsBasisByNiceBasis( B )  Category
  
  This  filter  indicates  that  the  basis  B  delegates  tasks  such  as the
  computation  of  coefficients  (see Coefficients  (61.6-3)) to a basis of an
  isomorphic nicer free left module.
  
  61.11-6 IsHandledByNiceBasis
  
  IsHandledByNiceBasis( M )  Category
  
  For  a  free  left module M in this category, essentially all operations are
  performed using a nicer free left module, which is usually a row module.
  
  
  61.12 How to Implement New Kinds of Vector Spaces
  
  61.12-1 DeclareHandlingByNiceBasis
  
  DeclareHandlingByNiceBasis( name, info )  function
  InstallHandlingByNiceBasis( name, record )  function
  
  These  functions  are used to implement a new kind of free left modules that
  shall be handled via the mechanism of nice bases (see 61.11).
  
  name  must  be a string, a filter f with this name is created, and a logical
  implication from f to IsHandledByNiceBasis (61.11-6) is installed.
  
  record must be a record with the following components.
  
  detect 
        a  function  of  four arguments R, l, V, and z, where V is a free left
        module over the ring R with generators the list or collection l, and z
        is  either  the  zero  element of V or false (then l is nonempty); the
        function  returns  true  if  V  shall  lie  in the filter f, and false
        otherwise;  the  return value may also be fail, which indicates that V
        is not to be handled via the mechanism of nice bases at all,
  
  NiceFreeLeftModuleInfo 
        the NiceFreeLeftModuleInfo method for left modules in f,
  
  NiceVector 
        the  NiceVector  method  for  left modules V in f; called with V and a
        vector  v ∈ V, this function returns the nice vector r associated with
        v, and
  
  UglyVector
        the UglyVector (61.11-2) method for left modules V in f; called with V
        and  a  vector  r  in the NiceFreeLeftModule value of V, this function
        returns the vector v ∈ V to which r is associated.
  
  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
  DeclareHandlingByNiceBasis  and InstallHandlingByNiceBasis, which causes the
  installation  of  the necessary methods and adds the pair [ f,record.detect]
  to   the   global   list  NiceBasisFiltersInfo.  The  LeftModuleByGenerators
  (57.1-10) methods call CheckForHandlingByNiceBasis (61.12-3), which sets the
  appropriate filter for the desired left module if applicable.
  
  61.12-2 NiceBasisFiltersInfo
  
  NiceBasisFiltersInfo  global variable
  
  An overview of all kinds of vector spaces that are currently handled by nice
  bases  is  given  by  the global list NiceBasisFiltersInfo. Examples of such
  vector  spaces  are  vector  spaces  of  field  elements (but not the fields
  themselves)  and  non-Gaussian  row  and  matrix spaces (see IsGaussianSpace
  (61.9-3)).
  
  61.12-3 CheckForHandlingByNiceBasis
  
  CheckForHandlingByNiceBasis( R, gens, M, zero )  function
  
  Whenever   a   free   left  module  is  constructed  for  which  the  filter
  IsHandledByNiceBasis  may  be  useful, CheckForHandlingByNiceBasis should be
  called.   (This   is   done  in  the  methods  for  VectorSpaceByGenerators,
  AlgebraByGenerators, IdealByGenerators etc. in the GAP library.)
  
  The  arguments of this function are the coefficient ring R, the list gens of
  generators, the constructed module M itself, and the zero element zero of M;
  if gens is nonempty then the zero value may also be false.
  
  
  61.13 Tensor Products and Exterior and Symmetric Powers
  
  61.13-1 TensorProduct
  
  TensorProduct( list )  operation
  TensorProduct( V, W, ... )  operation
  
  Here  list 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.
  
  In the second form, the vector spaces are given individually.
  
  Elements of the tensor product V_1⊗ ⋯ ⊗ V_k are linear combinations of v_1⊗⋯
  ⊗  v_k,  where  the v_i are arbitrary basis elements of V_i. In GAP a tensor
  element like that is printed as
  
    Example  
       v_1<x> ... <x>v_k
  
  
  Furthermore, the zero of a tensor product is printed as
  
    Example  
     <0-tensor>
  
  
  This  does  not  mean  that  all tensor products have the same zero element:
  zeros of different tensor products have different families.
  
    Example  
    gap> V:=TensorProduct(Rationals^2, Rationals^3);
    <vector space over Rationals, with 6 generators>
    gap> Basis(V);
    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 ]) ] )
  
  
  See also KroneckerProduct (24.5-8).
  
  61.13-2 ExteriorPower
  
  ExteriorPower( V, k )  operation
  
  Here V must be a vector space. This function returns the k-th exterior power
  of V.
  
  Elements  of  the  exterior power ⋀^k V are linear combinations of v_i_1∧⋯ ∧
  v_i_k,  where  the v_i_j are basis elements of V, and 1 ≤ i_1 < i_2 ⋯ < i_k.
  In GAP a wedge element like that is printed as
  
    Example  
       v_1/\ ... /\v_k
  
  
  Furthermore, the zero of an exterior power is printed as
  
    Example  
     <0-wedge>
  
  
  This  does  not  mean  that  all exterior powers have the same zero element:
  zeros of different exterior powers have different families.
  
    Example  
    gap> V:=ExteriorPower(Rationals^3, 2);
    <vector space of dimension 3 over Rationals>
    gap> Basis(V);
    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 ]) ] )
  
  
  61.13-3 SymmetricPower
  
  SymmetricPower( V, k )  operation
  
  Here  V  must  be  a  vector space. This function returns the k-th symmetric
  power of V.
  
    Example  
    gap> V:=SymmetricPower(Rationals^3, 2);
    <vector space over Rationals, with 6 generators>
    gap> Basis(V);
    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 ])
     ] )