= Solution
Because every column of $X$ has <sample mean> zero, every column $u_i$ of $U=XV$ also has sample mean zero. The <spectral theorem for real symmetric matrices> gives
$$
U^TU=V^TX^TXV=V^T\Sigma V=\Lambda.
$$
Hence for $i\ne j$ the <sample covariance> is $n^{-1}u_i^Tu_j=0$, while the <sample variance> of $u_i$ is
$$
\frac1n u_i^Tu_i=\frac{\Lambda_{ii}}n.
$$
\b[Thus the $u_i$ are pairwise uncorrelated <sample principal components>, ordered by decreasing sample variance.]
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