= Solution
{c}
The conditional mean estimator is the <posterior mean>
$$
\boxed{\widehat u_{\mathrm{CM}}(m)=\mathbb E[u\mid m]
=\int_{\mathbb R^d}u\pi^m(u)\,du,}
$$
provided the first <moment> is finite. A <maximum a posteriori estimate> is any maximizer of the <posterior density> relative to <Lebesgue measure>:
$$
\boxed{\widehat u_{\mathrm{MAP}}(m)\in\operatorname*{arg\,max}_{u\in\mathbb R^d}\pi^m(u).}
$$
Existence of the <posterior density> alone does not guarantee a finite <posterior mean> or existence of a maximizing point.
For the stated <multivariate normal distribution> and independent standard <Gaussian noise>, the negative log of the <posterior density>, up to a constant, is
$$
\frac12\|m-Au\|^2+\frac12u^T\Sigma^{-1}u.
$$
Its <precision matrix> $A^TA+\Sigma^{-1}$ is a <positive-definite matrix>, so <Gaussian conjugacy for a normal linear model> gives
$$
\boxed{\widehat u_{\mathrm{CM}}=(A^TA+\Sigma^{-1})^{-1}A^Tm
=\Sigma A^T(A\Sigma A^T+I_k)^{-1}m.}
$$
The two forms agree because
$$
(A^TA+\Sigma^{-1})\Sigma A^T=A^T(A\Sigma A^T+I_k).
$$
The <covariance matrix> is $(A^TA+\Sigma^{-1})^{-1}$, and the <maximum a posteriori estimate> equals the <posterior mean> for this <multivariate normal distribution>.
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