Find standard matrices for the following linear transformations
(1) $ T (\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}) = \begin{pmatrix} 2 \\ 2 \\2 \end{pmatrix}$ , $ \quad $ $ T (\begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}) = \begin{pmatrix} 1 \\ -1 \\ -1 \end{pmatrix}$, $ \quad $ $ T (\begin{pmatrix} 0 \\ 1 \\ -2 \end{pmatrix}) = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}$ , $ \quad $ $ \quad $$ M = \begin{pmatrix}1 & 1 & 0 \\ 2 & 1 & 1 \\ 3 & 2 & -2 \end{pmatrix}. $
$ \quad $ Then, $ T (\begin{pmatrix} x \\ y \\ z \end{pmatrix}) = \begin{pmatrix} 2 & 1 & 0 \\ 2 & -1 & 0 \\ 2 & -1 & 0 \end{pmatrix} M^{-1} \begin{pmatrix} x \\ y \\ z \end{pmatrix} $ ?
(2) $ \quad $ Let $ S_1 = \begin{pmatrix}3 & 2 & 1 \\ 1 & 2 & 3 \end{pmatrix}. $ be the standard matrix for linear transformation S. Let $ T_1 = \begin{pmatrix}1 & 2 \\ 2 & 1 \\ -1 & 1 \end{pmatrix}. $ be the standard matrix for linear transformation T.
$\quad $ Standard matrix for linear transformation S 0 T = $ S_1T_1 = \begin{pmatrix} 6 & 9 \\ 2 & 7 \end{pmatrix}$ ?
I am not sure if the standard matrices are correct. Is there an easier way to check if the standard matrices are correct without finding $ M^{-1} $ in (1)? Thank you.