= Solution
{c}
The new score columns are orthogonal, since $Z^TZ=V^TX^TXV$ is diagonal. This decouples the slope <normal equations>; with positive eigenvalues and the stated error <covariance>, the two estimated score coefficients are uncorrelated. It also makes variance explained and possible dimension reduction explicit.
The trade-off is interpretation: each <principal component> is a mixture of the original variables, so its regression coefficient is less directly connected to changing one original input. Keeping both components is an invertible reparameterization and gives \b[exactly the same fitted values and residual sum of squares]. It does not repair a zero <eigenvalue> or remove instability associated with a very small one. Dropping a low-variance component may improve numerical stability and reduce estimator variance, but changes the model and can discard predictive information, since <principal component analysis> uses predictor variance rather than the response. This is the distinction between a full rotation and <principal component regression> with truncation.
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