9.6 Base

Base( V )

Base computes a base of the given vector space V. The result is returned as a list of elements of the vector space V.

The base of a vector space is defined to be a minimal generating set. It can be shown that for a given vector space V each base has the same number of elements, which is called the dimension of V (see Dimension).

Unfortunately, no better algorithm is known to compute a base in general than to browse through the list of all elements of the vector space. So be careful when using this command on plain vector spaces.

    gap> f := GF(3);
    GF(3)
    gap> m1 := [[ f.one, f.one, f.zero, f.zero ]];
    [ [ Z(3)^0, Z(3)^0, 0*Z(3), 0*Z(3) ] ]
    gap> m2 := [[ f.one, f.one, f.one, f.zero ]]; 
    [ [ Z(3)^0, Z(3)^0, Z(3)^0, 0*Z(3) ] ]
    gap> V := VectorSpace( [ m1, m2, m1+m2 ], GF(3) );
    VectorSpace( [ [ [ Z(3)^0, Z(3)^0, 0*Z(3), 0*Z(3) ] ], 
      [ [ Z(3)^0, Z(3)^0, Z(3)^0, 0*Z(3) ] ], 
      [ [ Z(3), Z(3), Z(3)^0, 0*Z(3) ] ] ], GF(3) )
    gap> Base( V );
    [ [ [ Z(3)^0, Z(3)^0, 0*Z(3), 0*Z(3) ] ], 
      [ [ Z(3)^0, Z(3)^0, Z(3)^0, 0*Z(3) ] ] ]
    gap> Dimension( V );
    2 

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