Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 30 / 1 / d / Solution

Because both vectors are linear transformations of the same multivariate normal response, $(\hat{Y},e)$ is jointly multivariate normal. Its cross-covariance matrix isZero cross-covariance implies independence for jointly multivariate normal vectors, including singular ones. Therefore the fitted values and the entire vector of regression residuals are independent. The fitted-residual orthogonality identity gives the zero covariance; the normal distribution assumption is what upgrades it to independence.