Past exam of the numerical linear algebra course of the University of Oxford / 2025 / 1 / a / i / Solution

The orthonormal columns give $A^{T}A=I_n$. Thus $P=A{A}^T$ is the orthogonal projection onto the column space of $A$, andThe two terms are orthogonal, so the Pythagorean identity givesOnly the first term depends on $x$. Consequently the linear least-squares problem has the unique answer