Exercise 4

Consider the transformation $t\colon\mathcal{P}_{2}\to\mathcal{P}_{2}$ described by $x^{2}\mapsto x+1$, $x\mapsto x^{2}-1$, and $1\mapsto 3$.

• Find $T={\rm Rep}_{B,B}(t)$ where $B=\langle x^{2},x,1\rangle$.

• Find $\hat{T}={\rm Rep}_{D,D}(t)$ where $D=\langle 1,1+x,1+x+x^{2}\rangle$.

• Find the matrix $P$ such that $\hat{T}=PTP^{-1}$.