Let $f : \mathbb{R}^{3} \to \mathbb{R}^{3}$ be the linear transformation defined by $$f(\vec{x}) = \left[\begin{array}{ccc} -3 &1 &-3\cr 3 &5 &0\cr -3 &2 &-1 \end{array}\right] \vec{x}.$$ Let $$\begin{array}{lcl} \mathcal{B} & = & \lbrace \left<0,-1,1\right>, \left<0,0,1\right>, \left<-1,1,0\right> \rbrace, \\ \mathcal{C} & = & \lbrace \left<0,1,1\right>, \left<0,0,1\right>, \left<1,1,2\right> \rbrace, \end{array}$$ be two different bases for $\mathbb{R}^{3}$. Find the matrix $\lbrack f \rbrack_{\mathcal{B}}^{\mathcal{C}}$ for $f$ relative to the basis $\mathcal{B}$ in the domain and $\mathcal{C}$ in the codomain.

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