Exercise 6

Exhibit a nontrivial similarity relationship by letting $t\colon\mathbb{C}^{2}\to\mathbb{C}^{2}$ act in this way,

picking two bases $B,D$, and representing $t$ with respect to them, $\hat{T}={\rm Rep}_{B,B}(t)$ and $T={\rm Rep}_{D,D}(t)$. Then compute the $P$ and $P^{-1}$ to change bases from $B$ to $D$ and back again.