Let $B=\{(1,2), (-1,-1)}$ and $B^{\prime) =\{(-4,1),(0,2)\}$ be bases for $R^{2}$, and let $A=\left[\begin{array}{rr}0 & 2 1 -1 & 1\end{array} ight]$ be the matrix for $T: R^{2} ightarrow R^{2}$ relative to $B$. (a) Find the transition matrix $P$ from $B^{\prime) $ to $B$. (b) Use the matrices $P$ and $A$ to find $[\mathbf {v}l_{B} $ and $[T(\mathbf{v})}_{B}$, where \left[\begin{array}{1} \mathbf {v} \end{array} ight]_{B^{\prime} }=\left[\begin{array}{11} -5 & 2 \end{array} ight]^{T} $$ $$ (c) Find $P^{-1}$ and $A^{\prime) $ (the matrix for $T$ relative to $B^{\prime) $ ). (d) Find $[T(\mathbf {v})}_{B^{\prime}}$ two ways. $$ [T(\mathbf {v})}_{B^{\prime}}=P^{-1}[T(\mathbf {v})}_{B}=\left[\begin{array}{:} \square \square \Downarrow \mathbb{v} \end{array} ight] ightarrow $$ $$ [T(\mathbf{v})}_{B^{\prime} }=A^{\prime) [\mathbf {v}l_{B^{\prime} }=\left [\begin{array}{1} \square \square \end{array} ight] \Rightarrow $$ CS.JG. 129