Ok, so I have a question which is:
$ A = \left ( \begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \end{array} \right ) $
Find non-singular matrices P,Q such that $PAQ$ is a diagonal matrix of 1's and 0's, with the 1's appearing before the 0's, via a change of basis.
Now I can do this simply with row/column operations to get that: $P=I$ and $Q= \left ( \begin{array}{ccc} 1 & 0 & -1 \\ 0 & 1 & -1 \\ 0 & 0 & 1 \end{array} \right ) $
But I am unsure how to do this via a change of basis?
I can see that $null(A)=span(\{(1,1,-1)\})$ and $range(A)=span(\{(1,0,0),(0,1,0)\})$ and that a change of basis matrix to this basis is given by:
$ \left ( \begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & -1 \end{array} \right ) $
and I can see that this does the job for Q, but I'm not really sure what I have done here (I've just blindly changed basis and this matrix has popped out!)
any help as to how I should be going about this and what it is I am actually doing would be great.
Thanks very much for any help