= Solution
For a bipartite <density operator> $\rho_{SA}$, the <reduced density matrix> of $S$ is
$$
\boxed{\rho_S=\operatorname{Tr}_A\rho_{SA}}.
$$
It is characterized by
$$
\operatorname{Tr}_S(M_S\rho_S)
=\operatorname{Tr}_{SA}[(M_S\otimes I_A)\rho_{SA}]
$$
for every observable $M_S$ on $S$. Equivalently, in any <orthonormal basis> $\{|a_k\rangle\}$ of $A$, the <partial trace> is
$$
\rho_S=\sum_k(I_S\otimes\langle a_k|)\rho_{SA}
(I_S\otimes|a_k\rangle),
$$
and the result is independent of that basis.
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