Example

Let S=span{Transpose[1 0 1], Transpose[0 1 1], Transpose[1 1 2], Transpose[1 2 1], Transpose[-1 1 2]} be a subspace of R^3.

In[29]:=

  Clear[a,m]
  a={{1,0,1,1,-1},{0,1,1,2,1},{1,1,2,1,2}};
  MatrixForm[a]

Out[29]=

  1    0    1    1    -1
  
  0    1    1    2    1
  
  1    1    2    1    2

In[30]:=

  m=RowReduce[Transpose[a]];
  MatrixForm[m]

Out[30]=

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

In[31]:=

  MatrixForm[Transpose[m]]

Out[31]=

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

So, the standard basis for S is actually the standard unit vector basis for R^3, that is, S=R^3.

Up to Standard Basis of a Subspace S of R^n