58.3 Domains of Subspaces of Vector Spaces

Let V be a finite vector space. In the first form, Subspaces returns the domain of all subspaces of V. In the second form, k must be a nonnegative integer, and Subspaces returns the domain of all k-dimensional subspaces of V.

Special Size and Iterator methods are provided for these domains.

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)>

The domain of all subspaces of a (finite) vector space or of all subspaces of fixed dimension, as returned by Subspaces (see Subspaces) lies in the category IsSubspacesVectorSpace.

gap> V:= VectorSpace( GF(2^2), [ [ Z(2)^0, 0*Z(2) ], [ 0*Z(2), Z(2)^0 ] ] );;
gap> D:= Subspaces( V, 1 );
Subspaces( VectorSpace( GF(2^2), [ [ Z(2)^0, 0*Z(2) ], [ 0*Z(2), Z(2)^0 ] 
 ] ), 1 )
gap> IsSubspacesVectorSpace( D );
true

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