Rework Example CIVLT, only in place of the basis $B$ for $P_2$, choose instead to use the basis $C=\set{1,\,1+x,\,1+x+x^2}$. This will complicate writing a generic element of the domain of $\ltinverse{T}$ as a linear combination of the basis elements, and the algebra will be a bit messier, but in the end you should obtain the same formula for $\ltinverse{T}$. The inverse linear transformation is what it is, and the choice of a particular basis should not influence the outcome.