Past exam of the mathematics course of the University of Cambridge / 2020 / ib / Paper 2 / 18H / e / Solution

Full column rank of $X$ makes $X^T:{\mathbb{R}}^n\to {\mathbb{R}}^p$ surjective. Given $x\in {\mathbb{R}}^p$, choose $u$ with $x=X^{T}u$. The fitted values for the unrestricted and restricted regressions areThereforeUsing $\mathrm{cov}(Y)={\sigma }^2{I}_n$ and idempotence of the projections,Part (d) now gives