Chapter 4.7, Problem 37E

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

$B=\left\{\left(1,3\right),\left(-2,-2\right)\right\}$, $B^{\prime }=\left\{\left(-12,0\right),\left(-4,4\right)\right\}$,

${\left[x\right]}_{B^{\prime }}=\left[\begin{array}{c}-1\\ 3\end{array}\right]$