Answer
See below
Work Step by Step
Assume $A,B \in M_2(R)\\
c \in R$
Obtain $T(A+B)\\
=S^{-1}(A+B)S\\
=(S^{-1}A+S^{-1}B)S\\
=S^{-1}AS+S^{-1}BS\\
=T(A)+T(B)$
and $T(cA)\\
=S^{-1}(cA)S\\
=cS^{-1}AS\\
=cT(A)$
Hence, $T$ is a linear transformation
Obtain: $Ker(T)=\{A \in M_2(R):T (A)=0\}\\
=\{A \in M_2(R): S^{-1}AS=0\}\\
\rightarrow A=0\\
\rightarrow Ker(T)=\{0\}$
$T$ is one-to-one (1)
Apply Rank Nullity Theorem:
$\dim [Ker(T)]+\dim [Rng(T)]=\dim M_2(R)\\
0+\dim [Rng(T)]=4\\
\dim [Rng(T)]=4$
Since $\dim [Rng(T)]=4=\dim M_2(R)$,
hence, $T$ is onto (2)
From (1) and (2), $T$ is isomorphism.
