Left Nullspace of A

The Left Nullspace of A is just the nullspace of Transpose[A]. Use the above except use it on Transpose[A].

Consider A=Transpose[Transpose[A]]. If M is the GJSF of A, then M=D.A Partition D by rows into D=Transpose[D11 D21], where D11 corresponds to the nonzero rows of M and D21 corresponds to the zero rows of M. The columns of Transpose[D21] form the standard basis for the Left Nullspace of A.


  a={{1,2,1,2},{2,1,2,1},{3,2,3,2},{3,3,3,3},{5,3,5,3}};
  MatrixForm[a]


  aI=AppendRows[a,IdentityMatrix[5]];
  MatrixForm[aI]


  GJSFofaI=RowReduce[aI];
  MatrixForm[GJSFofaI]

Because this is large, let's get the last five columns; that will be D. Notice the TakeColumns command. The -5 means the last four columns.


  Clear[d]
  d=TakeColumns[GJSFofaI,-5];
  MatrixForm[d]

The last three rows correspond to the nonzero rows of M. The standard basis for the Left Nullspace of A is
{Transpose[1 0 0 -7/6 1/2], Transpose[0 1 0 1/6 -1/2],
Transpose[0 0 1 -1/6 -1/2]}.

Up to Bases for These Subspaces