For all $n \in \mathbb { N }$, let $G _ { n } = \left( \left( X ^ { i - 1 } \mid X ^ { j - 1 } \right) \right) _ { 1 \leqslant i , j \leqslant n + 1 }$ be the Gram matrix and let $\left( V _ { n } \right) _ { n \in \mathbb { N } }$ be an orthogonal system. Let $Q _ { n } = \left( q _ { i , j } \right) _ { 1 \leqslant i , j \leqslant n + 1 }$ be the matrix of the family $\left( V _ { 0 } , V _ { 1 } , \ldots , V _ { n } \right)$ in the basis $\left( 1 , X , \ldots , X ^ { n } \right)$ of $\mathbb { R } _ { n } [ X ]$. Show that $Q _ { n }$ is upper triangular and that $\operatorname { det } Q _ { n } = 1$.
For all $n \in \mathbb { N }$, let $G _ { n } = \left( \left( X ^ { i - 1 } \mid X ^ { j - 1 } \right) \right) _ { 1 \leqslant i , j \leqslant n + 1 }$ be the Gram matrix and let $\left( V _ { n } \right) _ { n \in \mathbb { N } }$ be an orthogonal system. Let $Q _ { n } = \left( q _ { i , j } \right) _ { 1 \leqslant i , j \leqslant n + 1 }$ be the matrix of the family $\left( V _ { 0 } , V _ { 1 } , \ldots , V _ { n } \right)$ in the basis $\left( 1 , X , \ldots , X ^ { n } \right)$ of $\mathbb { R } _ { n } [ X ]$. Show that $Q _ { n }$ is upper triangular and that $\operatorname { det } Q _ { n } = 1$.