= Solution
Since $X=UV^T$ and $X^TX=V\Lambda V^T$,
$$
\widehat Y_{\rm MLE}
=X(X^TX)^{-1}X^TY
=U\Lambda^{-1}U^TY.
$$
Therefore
$$
\boxed{\widehat Y_{\rm MLE}=U\Lambda^{-1}U^TY.}
$$
Put $w_i=u_i/\sqrt{\Lambda_{ii}}$. By part (c), the $w_i$ are <normalized sample principal components> forming an <orthonormal set>, and
$$
U\Lambda^{-1}U^T=\sum_{i=1}^p w_iw_i^T.
$$
This is the <orthogonal projection onto a finite-dimensional subspace> spanned by the $w_i$, so $\widehat Y_{\rm MLE}$ is the unique closest point in that subspace to $Y$.
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