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

Because $X$ has full column rank, $X^{T}X$ is invertible. The ordinary least squares estimator isIt satisfies the normal equationsso the residual is orthogonal to the column space of $X$. For any $\beta$,The two terms are orthogonal, and the Pythagorean theorem in an inner-product space givesThus $\hat{\beta }$ minimizes the least-squares objective.