Row Space of A

The rows of the GJSF of A form the standard basis for the Row Space of A

In[41]:=

  MatrixForm[a]
  MatrixForm[RowReduce[a]]

Out[41]=

  1   2   1   2
  
  2   1   2   1
  
  3   2   3   2
  
  3   3   3   3
  
  5   3   5   3

Out[42]=

  1   0   1   0
  
  0   1   0   1
  
  0   0   0   0
  
  0   0   0   0
  
  0   0   0   0

The standard basis for the Row Space of A is
{ [1 0 1 0], [0 1 0 1]}.