Write the matrix $H$ of the inner product $\phi(P,Q) = \int_0^1 P(t)Q(t)\,\mathrm{d}t$ in the canonical basis of $\mathbb{R}_{n-1}[X]$, that is, the matrix with general term $h_{i,j} = \phi\left(X^i, X^j\right)$ where the indices $i$ and $j$ vary between 0 and $n-1$.