Find a transition matrix from S to B.

Question

I was asked this for the previous question: Find the coordinates of $ [u]_s = ( 1,2,3) $ relative to the basis $ B = {(0,2,1), (-2,1,0),(1,1,1)} $. Next I need to find a transition matrix I have set this up as [B|S]. In the text it says to turn this into the form $ [I_3|p^{-1}] $ by gauss jordan elimination. So then should I use row reduction on B and should S be written as $$ \begin{matrix} 1 & 2 & 3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{matrix} $$

Answer 1

You want to solve $$ u = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} = u_1 b_1 + u_2 b_2 + u_3 b_3 = \begin{pmatrix} 0 & -2 & 1 \\ 2 & 1 & 1 \\ 1 & 0 & 1 \end{pmatrix} \begin{pmatrix} u_1 \\ u_2 \\ u_3 \end{pmatrix} $$ The bit about the transition matrix I do not get.

Doing Gauss-Jordan and ending up with $[I\mid p^{-1}]$ is what you can do if you want to invert a matrix $p$.