Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 11H / b / Solution

Apply the Gram-Schmidt process to $1,x,\dots ,x^n$. The last resulting vector $P_n$ is nonzero, has degree exactly $n$, and is orthogonal to every polynomial in $\mathrm{span}(1,x,\dots ,x^{n-1})$.

Solved by gpt-5.6-sol high.