Given v$_1=\left( \begin{array}{ccc} 3\\ -4 \end{array} \right)$, v$_2=\left( \begin{array}{ccc} 2\\ 5 \end{array} \right)$, $S=\left( \begin{array}{ccc} -1 & 7\\ 2 &-5 \end{array} \right)$
Find the vectors u$_1$, u$_2$ such that $S$ will be a transition matrix from [v$_1$, v$_2$] -> [u$_1$, u$_2$]
So transition matrices work like this: $V = U*S$
Then $V*S^{-1}=U$
Where $V=\left( \begin{array}{ccc} 3 & 2\\ -4 & 5\end{array} \right)$ and $S^{-1} = \frac{1}{-9}\left( \begin{array}{ccc} 5 &-7\\ -2 & 1 \end{array} \right)=\left( \begin{array}{ccc} -5/9 &7/9\\ 2/9 & -1/9 \end{array} \right)$
Then $V*S^{-1} = \left( \begin{array}{ccc} -11/9 &19/9\\ 30/9 & -33/9 \end{array} \right)$ Which is supposed to be $U$ but it's wrong if you check the answer by plugging it into $V=U*S$
What am I doing incorrectly?
Note that the inverse should be $$S^{-1} = \left( \begin{array}{ccc} 5/9 &7/9\\ 2/9 & 1/9 \end{array} \right)$$