Basic Examples 
(3)
 

The sum of a 2D subspace and a 3D subspace of
4

:
In[602]:=
[◼]
SumBasis
[{{1,1,1,1},{2,2,3,4}},{{2,3,4,7},{1,0,1,0},{0,1,0,3}}]
Out[602]=
{{1,1,1,1},{2,2,3,4},{2,3,4,7},{0,1,0,3}}
———
The sum of a subspace and a subspace of the subspace:
In[603]:=
[◼]
SumBasis
[{{1,1,1,1},{2,2,3,4}},{{3,3,5,7}}]
Out[603]=
{{1,1,1,1},{2,2,3,4}}
———
A sum of three subspaces:
In[604]:=
[◼]
SumBasis
[{{1,1,1,1,1},{1,2,3,4,5}},{{1,0,1,0,1},{0,1,0,3,1}},{{1,1,1,3,2},{2,3,4,5,6}}]
Out[604]=
{{1,1,1,1,1},{1,2,3,4,5},{1,0,1,0,1},{0,1,0,3,1}}