In linear algebra, I remember that there was something special about the submatrix in the top right of an rref'd matrix.
1 0 0 | 0 1 0 1 0 | -1 0 0 0 1 | 1 0 ------------- 0 0 0 0 0
You would ammend an identity matrix (in this case $2\times2$) above the submatrix and get two vectors in 5 dimensions. I think they say something about the kernel or so, but I do not really remember.
What do those two vectors tell me about the original matrix?