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Since $X=U{V}^T$ and $X^{T}X=V\Lambda V^T$,ThereforePut $w_i=u_i/\sqrt{{\Lambda }_{ii}}$. By part (c), the $w_i$ are normalized sample principal components forming an orthonormal set, andThis is the orthogonal projection onto a finite-dimensional subspace spanned by the $w_i$, so ${\hat{Y}}_{\mathrm{MLE}}$ is the unique closest point in that subspace to $Y$.

For $X^{T}X=V\Lambda V^T$, ridge regression multiplies the fitted response along the $i$th normalized sample principal component by the shrinkage factor ${\Lambda }_{ii}/({\Lambda }_{ii}+\lambda )$. Directions with variance much larger than $\lambda$ are nearly retained, while directions with variance much smaller than $\lambda$ are nearly removed.