# 33.7 Operations for Row Spaces

Comparisons of Row Spaces

`V = W` :

returns `true` if the two row spaces V, W are equal as sets, and `false` otherwise.

`V < W` :

returns `true` if the row space V is smaller than the row space W, and `false` otherwise. The first criteria of this ordering are the comparison of the fields and the dimensions, row spaces over the same field and of same dimension are compared by comparison of the reversed canonical bases (see CanonicalBasis).

Arithmetic Operations for Row Spaces

`V + W` :

returns the sum of the row spaces V and W, that is, the row space generated by V and W. This is computed using the Zassenhaus algorithm.

`V / U` :

returns the quotient space of V modulo its subspace U (see Quotient Spaces).

```    gap> v:= GF(2)^2; v.name:= "v";;
RowSpace( GF(2), [ [ Z(2)^0, 0*Z(2) ], [ 0*Z(2), Z(2)^0 ] ] )
gap> s:= Subspace( v, [ [ 1, 1 ] * Z(2) ] );
Subspace( v, [ [ Z(2)^0, Z(2)^0 ] ] )
gap> t:= Subspace( v, [ [ 0, 1 ] * Z(2) ] );
Subspace( v, [ [ 0*Z(2), Z(2)^0 ] ] )
gap> s = t;
false
gap> s < t;
false
gap> t < s;
true
gap> u:= s+t;
Subspace( v, [ [ Z(2)^0, Z(2)^0 ], [ 0*Z(2), Z(2)^0 ] ] )
gap> u = v;
true
gap> f:= u / s;
Subspace( v, [ [ Z(2)^0, Z(2)^0 ], [ 0*Z(2), Z(2)^0 ] ] ) /
[ [ Z(2)^0, Z(2)^0 ] ] ```

GAP 3.4.4
April 1997