Chapter 6.4, Problem 18E

Repeat Exercise 17 for

$\begin{array}{l}B=\left\{\left(1,1,-1\right),\left(1,-1,1\right),\left(-1,1,1\right)\right\}\\ B^{\prime }=\left\{\left(1,0,0\right),\left(0,1,0\right),\left(0,0,1\right)\right\},\text{ and}\end{array}$

${\left[\text{v}\right]}_{B^{\prime }}={\left[\begin{array}{ccc}2& 1& 1\end{array}\right]}^T$.

(Use matrix A in Exercise 17.)

Let $B=\left\{\left(1,1,0\right),\left(1,0,1\right),\left(0,1,1\right)\right\}$ and $B^{\prime }=\left\{\left(1,0,0\right),\left(0,1,0\right),\left(0,0,1\right)\right\}$ be bases for $R^3$, and let

$A=\left[\begin{array}{ccc}\frac{3}{2}& -1& -\frac{1}{2}\\ -\frac{1}{2}& 2& \frac{1}{2}\\ \frac{1}{2}& 1& \frac{5}{2}\end{array}\right]$

be the matrix for $T:R^3\to R^3$ relative to B.

(a) Find the transition matrix P from $B^{\prime }$ to B.

(b) Use the matrices P and A to find ${\left[v\right]}_B$ and ${\left[T\left(v\right)\right]}_{B^{\prime }}$, where ${\left[v\right]}_{B^{\prime }}={\left[\begin{array}{ccc}1& 0& -1\end{array}\right]}^T$.

(c) Find $P^{-1}$ and $A^{\prime }$ (the matrix for $T$ relative to $B^{\prime }$).

(d) Find ${\left[T\left(v\right)\right]}_{B^{\prime }}$ two ways.