Chapter 4.CR, Problem 70CR

Finding Transition and Coordinate Matrices In Exercises $69-72$, (a) find the transition matrix from $B$ to $B^{\prime }$, (b) find the two transition matrix from $B^{\prime }$ to $B$, (c) verify that the transition matrices are inverses of each other, and (d) Find the coordinate matrix ${\left[x\right]}_{B^{\prime }}$, given the coordinate matrix ${\left[x\right]}_B$.

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

${\left[x\right]}_B={\left[\begin{array}{cc}2& -2\end{array}\right]}^T$