<s>[INST] Change of Basis Given 2 Vectors and Transition Matrix 
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?
 [/INST] Note that the inverse should be
$$S^{-1} = \left( \begin{array}{ccc}
5/9 &7/9\\
2/9 & 1/9 \end{array} \right)$$
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