Goto Chapter: Top 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 Bib Ind

61 Vector Spaces

61.9 Row and Matrix Spaces

61.9-1 IsRowSpace

61.9-2 IsMatrixSpace

61.9-3 IsGaussianSpace

61.9-4 FullRowSpace

61.9-5 FullMatrixSpace

61.9-6 DimensionOfVectors

61.9-7 IsSemiEchelonized

61.9-8 SemiEchelonBasis

61.9-9 IsCanonicalBasisFullRowModule

61.9-10 IsCanonicalBasisFullMatrixModule

61.9-11 NormedRowVectors

61.9-12 SiftedVector

61.9-1 IsRowSpace

61.9-2 IsMatrixSpace

61.9-3 IsGaussianSpace

61.9-4 FullRowSpace

61.9-5 FullMatrixSpace

61.9-6 DimensionOfVectors

61.9-7 IsSemiEchelonized

61.9-8 SemiEchelonBasis

61.9-9 IsCanonicalBasisFullRowModule

61.9-10 IsCanonicalBasisFullMatrixModule

61.9-11 NormedRowVectors

61.9-12 SiftedVector

`‣ 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.

`‣ 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.

gap> V:= VectorSpace( Rationals, [ [ 1, 2, 3 ], [ 1, 1, 1 ] ] ); <vector space over Rationals, with 2 generators>

`‣ 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`.

gap> V:= VectorSpace( Rationals, [ [ 1, 2, 3 ], [ 1, 1, 1 ] ] );; gap> W:= Subspace( V, [ [ 0, 1, 2 ] ] ); <vector space over Rationals, with 1 generators>

`‣ 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.

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

`‣ 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.

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 generators> gap> Parent( U ) = V; true gap> AsSubspace( V, [ [ 1, 1, 1 ] ] ); fail

`‣ 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.

gap> GeneratorsOfVectorSpace( FullRowSpace( Rationals, 3 ) ); [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ]

`‣ TrivialSubspace` ( V ) | ( attribute ) |

For a vector space `V`, `TrivialSubspace`

returns the subspace of `V` that consists of the zero vector in `V`.

gap> V:= GF(3)^3;; gap> triv:= TrivialSubspace( V ); <vector space over GF(3), with 0 generators> gap> AsSet( triv ); [ [ 0*Z(3), 0*Z(3), 0*Z(3) ] ]

`‣ 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.

`‣ 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`

.

gap> D:= Subspaces( GF(3)^3 ); Subspaces( ( GF(3)^3 ) ) gap> Size( D ); 28 gap> iter:= Iterator( D );; gap> NextIterator( iter ); <vector space over GF(3), with 0 generators> gap> NextIterator( iter ); <vector space of dimension 1 over GF(3)> gap> IsSubspacesVectorSpace( D ); true

In **GAP**, a *basis* of a free left \(F\)-module \(V\) is a list of vectors \(B = [ v_1, v_2, \ldots, v_n ]\) 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 \(B[i]\).

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( `

is a semi-echelonized basis (see `V` )`IsSemiEchelonized`

(61.9-7)) that uses Gaussian elimination; such a basis is of the third kind. `Basis( `

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.`V`, `vectors` )

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.

`‣ 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.)

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

`‣ 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).

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

`‣ 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`

(78.2-1)) 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 78.2 and 78.3) being `CANONICAL_BASIS_FLAGS`

.

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 ] ]

`‣ 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.

`‣ BasisVectors` ( B ) | ( attribute ) |

For a vector space basis `B`, `BasisVectors`

returns the list of basis vectors of `B`. The lists `B` and `BasisVectors( `

are equal; the main purpose of `B` )`BasisVectors`

is to provide access to a list of vectors that does *not* know about an underlying vector space.

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 ] ]

`‣ UnderlyingLeftModule` ( B ) | ( attribute ) |

For a basis `B` of a free left module \(V\), say, `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.

gap> B:= Basis( GF(2)^6 );; UnderlyingLeftModule( B ); ( GF(2)^6 )

`‣ 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( `

is the list of coefficients of `B`, `v` )`v` w.r.t. `B` if `v` lies in \(V\), and `fail`

otherwise.

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

`‣ LinearCombination` ( B, coeff ) | ( operation ) |

If `B` is a basis object (see `IsBasis`

(61.5-1)) or a homogeneous list of length \(n\), say, and `coeff` is a row vector of the same length, `LinearCombination`

returns the vector \(\sum_{{i = 1}}^n \textit{coeff}[i] * \textit{B}[i]\).

Perhaps the most important usage is the case where `B` forms a basis.

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 ]

`‣ EnumeratorByBasis` ( B ) | ( attribute ) |

For a basis `B` of the free left \(F\)-module \(V\) of dimension \(n\), say, `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.

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) ] ]

`‣ IteratorByBasis` ( B ) | ( operation ) |

For a basis `B` of the free left \(F\)-module \(V\) of dimension \(n\), say, `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.

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) ]

`‣ 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.

`‣ IsIntegralBasis` ( B ) | ( property ) |

Let `B` be an \(S\)-basis of a *field* \(F\), say, 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.

`‣ IsNormalBasis` ( B ) | ( property ) |

Let `B` be an \(S\)-basis of a *field* \(F\), say, 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.

gap> B:= CanonicalBasis( GaussianRationals ); CanonicalBasis( GaussianRationals ) gap> IsIntegralBasis( B ); IsNormalBasis( B ); true false

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.

`‣ IsMutableBasis` ( MB ) | ( category ) |

Every mutable basis lies in the category `IsMutableBasis`

.

`‣ 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!

gap> MB:= MutableBasis( Rationals, [ [ 1, 2, 3 ], [ 0, 1, 0 ] ] ); <mutable basis over Rationals, 2 vectors>

`‣ 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.

gap> MB:= MutableBasis( Rationals, [ [ 1, 1], [ 2, 2 ] ] );; gap> NrBasisVectors( MB ); 1

`‣ ImmutableBasis` ( MB[, V] ) | ( operation ) |

`ImmutableBasis`

returns the immutable basis \(B\), say, 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.

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>

`‣ IsContainedInSpan` ( MB, v ) | ( operation ) |

For a mutable basis `MB` over the coefficient ring \(R\), say, and a vector `v`, `IsContainedInSpan`

returns `true`

is `v` lies in the \(R\)-span of the current basis vectors of `MB`, and `false`

otherwise.

`‣ CloseMutableBasis` ( MB, v ) | ( operation ) |

For a mutable basis `MB` over the coefficient ring \(R\), say, 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)).

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 ] ); gap> MB; <mutable basis over Rationals, 3 vectors> gap> IsContainedInSpan( MB, [ 1, 0, 0 ] ); true

`‣ IsRowSpace` ( V ) | ( filter ) |

A *row space* in **GAP** is a vector space that consists of row vectors (see Chapter 23).

`‣ IsMatrixSpace` ( V ) | ( filter ) |

A *matrix space* in **GAP** is a vector space that consists of matrices (see Chapter 24).

`‣ 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 the field \(F\), say, 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.

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

`‣ 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`.

gap> FullRowSpace( GF( 9 ), 3 ); ( GF(3^2)^3 ) gap> GF(9)^3; # the same as above ( GF(3^2)^3 )

`‣ 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``]`

.

gap> FullMatrixSpace( GF(2), 4, 5 ); ( GF(2)^[ 4, 5 ] ) gap> GF(2)^[ 4, 5 ]; # the same as above ( GF(2)^[ 4, 5 ] )

`‣ 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`.

gap> DimensionOfVectors( GF(2)^5 ); 5 gap> DimensionOfVectors( GF(2)^[2,3] ); [ 2, 3 ]

`‣ IsSemiEchelonized` ( B ) | ( property ) |

Let `B` be a basis of a Gaussian row or matrix space \(V\), say (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)).

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

`‣ 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.

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

`‣ 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.

`‣ 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.

`‣ 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)).

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) ])

`‣ 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.

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 ]

*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.

`‣ LeftModuleGeneralMappingByImages` ( V, W, gens, imgs ) | ( operation ) |

Let `V` and `W` be two left modules over the same left acting domain \(R\), say, and `gens` and `imgs` lists (of the same length) of elements in `V` and `W`, respectively. `LeftModuleGeneralMappingByImages`

returns the general mapping with source `V` and range `W` that is defined by mapping the elements in `gens` to the corresponding elements in `imgs`, and taking the \(R\)-linear closure.

`gens` need not generate `V` as a left \(R\)-module, and if the specification does not define a linear mapping then the result will be multi-valued; hence in general it is not a mapping (see `IsMapping`

(32.3-3)).

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

`‣ 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 \(R\), say, and `gens` and `imgs` lists (of the same length) of elements in `V` and `W`, respectively. `LeftModuleHomomorphismByImages`

returns the left \(R\)-module homomorphism with source `V` and range `W` that is defined by mapping the elements in `gens` to the corresponding elements in `imgs`.

If `gens` does not generate `V` or if the homomorphism does not exist (i.e., if mapping the generators describes only a multi-valued mapping) then `fail`

is returned. For creating a possibly multi-valued mapping from `V` to `W` that respects addition, multiplication, and scalar multiplication, `LeftModuleGeneralMappingByImages`

(61.10-1) can be used.

`LeftModuleHomomorphismByImagesNC`

does the same as `LeftModuleHomomorphismByImages`

, except that it omits all checks.

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 ]

`‣ 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`.

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>>

`‣ 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`

gap> V:= Rationals^3;; gap> W:= VectorSpace( Rationals, [ [ 1, 1, 1 ] ] );; gap> f:= NaturalHomomorphismBySubspace( V, W ); <linear mapping by matrix, ( Rationals^3 ) -> ( Rationals^2 )>

`‣ 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`.

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

`‣ 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`.

gap> A:= End( Rationals, Rationals^2 ); End( Rationals, ( Rationals^2 ) ) gap> Dimension( A ); 4

`‣ 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.

gap> V:= Rationals^2;; gap> W:= VectorSpace( Rationals, [ [ 1, 0, 1 ], [ 1, 2, 3 ] ] );; gap> H:= Hom( Rationals, V, W );; gap> IsFullHomModule( H ); true

`‣ 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.

gap> IsPseudoCanonicalBasisFullHomModule( Basis( H ) ); true

`‣ 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 \(v \in \textit{V}\) is defined as the matrix of the \(F\)-linear mapping \(v\) w.r.t. the bases \(B_S\) and \(B_R\).

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 \(B\), say, of \(V\) to the corresponding basis \(C\), say, 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).

`‣ 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.

`‣ 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( `

is either `v` )`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( `

is either `r` )`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( `

. If `V`, NiceVector( `V`, `v` ) ) = `v``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`.)

`‣ 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).

`‣ 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( `

is obtained as `B` )`Basis( NiceFreeLeftModule( `

. If basis vectors are stored then `V` ) )`NiceBasis( `

is the result of the call of `B` )`Basis`

with arguments `NiceFreeLeftModule( `

and the `V` )`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).

`‣ 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.

`‣ 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.

`‣ 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 \(v \in V\), 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 \(v \in V\) 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 \([ f, \)

\( ]\) to the global list `record`.detect`NiceBasisFiltersInfo`

. The `LeftModuleByGenerators`

(57.1-10) methods call `CheckForHandlingByNiceBasis`

(61.12-3), which sets the appropriate filter for the desired left module if applicable.

`‣ 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)).

`‣ 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`

.

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