Chapter 4.7, Problem 40E

Finding Transition and Coordinate Matrices In Exercises $37-40$, (a) find the transition matrix from $B$ to $B^{\prime }$, (b) find the transition matrix from $B^{\prime }$ to $B$, (c) verify that the two transition matrices are inverses of each other, and (d) find the coordinate matrix ${\left[x\right]}_B$, given the coordinate matrix ${\left[x\right]}_{B^{\prime }}$.

$\begin{array}{l}B=\left\{\left(1,1,1\right),\left(1,-1,1\right),\left(0,0,1\right)\right\},\\ B^{\prime }=\left\{\left(2,2,0\right),\left(0,1,1\right),\left(1,0,1\right)\right\},\\ {\left[x\right]}_{B^{\prime }}=\left[\begin{array}{c}2\\ 3\\ 1\end{array}\right]\end{array}$