33.18 SemiEchelonBasis

SemiEchelonBasis( V )
SemiEchelonBasis( V, vectors )

returns a semi-echelonized basis of the row space V. A basis is called semi-echelonized if the first non-zero element in every row is one, and all entries exactly below these elements are zero.

If a second argument vectors is given, these vectors are taken as basis vectors. Note that if the rows of vectors do not form a semi-echelonized basis then an error is signalled.

    gap> v:= GF(2)^2; v.name:= "v";;
    RowSpace( GF(2), [ [ Z(2)^0, 0*Z(2) ], [ 0*Z(2), Z(2)^0 ] ] )
    gap> SemiEchelonBasis( v );
    SemiEchelonBasis( v, [ [ Z(2)^0, 0*Z(2) ], [ 0*Z(2), Z(2)^0 ] ] )
    gap> b:= Basis( v, [ [ 1, 1 ], [ 0, 1 ] ] * Z(2) );
    Basis( v, [ [ Z(2)^0, Z(2)^0 ], [ 0*Z(2), Z(2)^0 ] ] )
    gap> IsSemiEchelonBasis( b );
    true
    gap> b;
    SemiEchelonBasis( v, [ [ 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 ] 

Previous Up Top Next
Index