# 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 ```

GAP 3.4.4
April 1997