Chapter 6.4, Problem 14E

Repeat Exercise 13 for $B=\left\{\left(1,1\right),\left(-2,3\right)\right\}$, $B^{\prime }=\left\{\left(1,-1\right),\left(0,1\right)\right\}$, and ${\left[v\right]}_{B^{\prime }}={\left[\begin{array}{cc}1& -3\end{array}\right]}^T$.

(Use matrix A in Exercise 13.)

Let $B=\left\{\left(1,3\right),\left(-2,-2\right)\right\}$ and $B^{\prime }=\left\{\left(-12,0\right),\left(-4,4\right)\right\}$ be bases for $R^2$, and let

$A=\left[\begin{array}{cc}3& 2\\ 0& 4\end{array}\right]$

be the matrix for $T:R^2\to R^2$ 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 ${\left[v\right]}_{B^{\prime }}={\left[\begin{array}{cc}-1& 2\end{array}\right]}^T$.

(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.