Determine values of some constant which equate linear operators whose linear transformation is through a different basis of the same vector space.
Let $S$ and $T$ be bases for a 2-dim vector space $V$ and let $A$ and $B$ be operators on $V$. Suppose that $[A]_S = \begin{pmatrix} 5 & c \\ c & -1 \end{pmatrix} $ and $[B]_T = \begin{pmatrix} -3 & 0 \\ 0 & 7 \end{pmatrix} $. Determine all values for $c$ such that $A = B$.
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