Chapter 4.7, Problem 41E

To determine

(a)

To find:

The transition matrix from $B$ to $B^{\prime }$ where $B=\left\{\left(4,2,-4\right),\left(6,-5,-6\right),\left(2,-1,8\right)\right\}$ and $B^{\prime }=\left\{\left(1,0,4\right),\left(4,2,8\right),\left(2,5,-2\right)\right\}$.

To determine

(b)

To find:

The transition matrix from $B^{\prime }$ to $B$ where $B=\left\{\left(4,2,-4\right),\left(6,-5,-6\right),\left(2,-1,8\right)\right\}$ and $B^{\prime }=\left\{\left(1,0,4\right),\left(4,2,8\right),\left(2,5,-2\right)\right\}$.

To determine

(c)

To verify:

The two transition matrices are inverses of each other.

To determine

(d)

To find:

The value of coordinate matrix ${\left[x\right]}_B$ where ${\left[x\right]}_{B^{\prime }}={\left[\begin{array}{ccc}1& -1& 2\end{array}\right]}^T$.