Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 102 / 1 / b / ii

Let $0\ne U\subseteq W$ be a subrepresentation and choosewith finite support. The $h$-eigenvalues $a+2r$ are pairwise distinct. By Lagrange interpolation, there is a polynomial $P$ which is one at one chosen eigenvalue appearing in $u$ and zero at all the others. Then $P(h)u$ is a nonzero scalar multiple of one basis vector $w_i$. Since $U$ is invariant under $h$, it contains $w_i$.