I have the following:
$ C = ($$ \left[ \begin{array}{cc} 1\\ -1 \end{array} \right], \left[ \begin{array} -1\\ 2 \end{array} \right]) , B = ($$ \left[ \begin{array}{cc} 2\\ 1 \end{array} \right], \left[ \begin{array}{c} 3\\ 2 \end{array} \right]) $ are bases of $\mathbb{R}^2$ ( no need to prove).
$ T:\mathbb{R}^2 \to \mathbb{R}^2 $ is linear transformation such that $ [T]^B _B = $$ \left[ \begin{array}{cc} 1&2\\ -1&1 \end{array} \right] $$ $
I need to calculate $ [T]^C _C$.
from $[T] ^B _B$ I can gather that:
$ T($$ \left[ \begin{array}{cc} 2\\ 1 \end{array} \right]) = 1 * $$ \left[ \begin{array}{cc} 2\\ 1 \end{array} \right] - 1* $$ \left[ \begin{array}{cc} 3\\ 2 \end{array} \right] $
$ T($$ \left[ \begin{array}{cc} 3\\ 2 \end{array} \right]) = 2 * $$ \left[ \begin{array}{cc} 2\\ 1 \end{array} \right] + 1* $$ \left[ \begin{array}{cc} 3\\ 2 \end{array} \right] $
And then I can know the values of those, but how to progress from here to find $[T]^C _C$?