Let $D$ denote the set of all $2\times2$ diagonal matrices. Consider an ordered basis $$β=\left\{\begin{bmatrix}1&0\\0&0\end{bmatrix}, \begin{bmatrix}0&0\\0&2\end{bmatrix}\right\}.$$
$T:D\to R^{2}$ is defined as $$T\left(\begin{bmatrix}a&0\\0&b\end{bmatrix}\right)=(a+b, 4a-5b)$$
Let Matrix $A$ be matrix representation of $T$ w.r.t $β$ for domain, and standard ordered basis for codomain.
Let Matrix $B$ be matrix representation of $T$ w.r.t $β$ for domain, and ordered basis $\{(1,-1), (1,1)\}$ for codomain.
Prove that $A$ is similar to $B$.
I have calculated $A$ and $B$, but unable to find any $P$ such that $B=P^{-1}AP$.