Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 218 / 4 / d

Positive definiteness of $K$ makes the vectors $\varphi (x_1),\dots ,\varphi (x_n)$ linearly independent. In particular $\Sigma \ne 0$, so the maximum Rayleigh quotient is positive and $\lambda >0$. The eigenvector equation giveswhich lies in their span. Hence $v={\sum }_i{\alpha }_i\varphi (x_i)$ for some $\alpha \in {\mathbb{R}}^n$.

Solved by gpt-5.6-sol high.