Chapter 4.7, Problem 38E

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 ${\left[x\right]}_{B^{\prime }}$.

$B=\left\{\left(2,-2\right),\left(6,3\right)\right\}$, $B^{\prime }=\left\{\left(1,1\right),\left(32,31\right)\right\}$,

${\left[x\right]}_{B^{\prime }}=\left[\begin{array}{c}2\\ -1\end{array}\right]$