Let $\mathcal{P}_{n}$ be the vector space of all polynomials of degree $n$ or less in the variable $x$. Let $D : \mathcal{P}_{3} \to \mathcal{P}_{2}$ be the linear transformation defined by $D(p(x)) = p'(x)$. That is, $D$ is the derivative operator. Let $$\begin{array}{lcl} \mathcal{B} & = & \lbrace 1, x, x^{2}, x^{3} \rbrace, \\ \mathcal{C} & = & \lbrace -1+x-x^{2}, -1+2x-x^{2}, 2-2x+x^{2} \rbrace, \end{array}$$ be ordered bases for $\mathcal{P}_{3}$ and $\mathcal{P}_{2}$, respectively. Find the matrix $\lbrack D \rbrack_{\mathcal{B}}^{\mathcal{C}}$ for $D$ relative to the basis $\mathcal{B}$ in the domain and $\mathcal{C}$ in the codomain.

$\lbrack D \rbrack_{\mathcal{B}}^{\mathcal{C}} =$ $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$