= Solution
Let $V=(X_1(t_1),X_1(t_2))^T$. The <Karhunen–Loève expansion> and Gaussianity give
$$
\operatorname{Cov}(a_{1k},V)
=\lambda_k(\phi_k(t_1),\phi_k(t_2))
$$
and
$$
\operatorname{Cov}(V)=
\Sigma=
\begin{pmatrix}
c_X(t_1,t_1)&c_X(t_1,t_2)\\
c_X(t_2,t_1)&c_X(t_2,t_2)
\end{pmatrix}.
$$
The <conditional multivariate normal distribution> formula yields
$$
\boxed{
\mathbb E[a_{1k}\mid X_1(t_1),X_1(t_2)]
=\lambda_k(\phi_k(t_1),\phi_k(t_2))
\Sigma^{-1}
\begin{pmatrix}X_1(t_1)\\X_1(t_2)\end{pmatrix}}.
$$
If $\Sigma$ is singular, the same formula uses its Moore-Penrose pseudoinverse.
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