For all $n \in \mathbb { N }$, let $G _ { n } = \left( \left( X ^ { i - 1 } \mid X ^ { j - 1 } \right) \right) _ { 1 \leqslant i , j \leqslant n + 1 }$, let $G _ { n } ^ { \prime } = \left( \left( V _ { i - 1 } \mid V _ { j - 1 } \right) \right) _ { 1 \leqslant i , j \leqslant n + 1 }$, and let $Q _ { n }$ be the matrix of the family $\left( V _ { 0 } , V _ { 1 } , \ldots , V _ { n } \right)$ in the basis $\left( 1 , X , \ldots , X ^ { n } \right)$. Show that $Q _ { n } ^ { \top } G _ { n } Q _ { n } = G _ { n } ^ { \prime }$, where $Q _ { n } ^ { \top }$ is the transpose of the matrix $Q _ { n }$.
For all $n \in \mathbb { N }$, let $G _ { n } = \left( \left( X ^ { i - 1 } \mid X ^ { j - 1 } \right) \right) _ { 1 \leqslant i , j \leqslant n + 1 }$, let $G _ { n } ^ { \prime } = \left( \left( V _ { i - 1 } \mid V _ { j - 1 } \right) \right) _ { 1 \leqslant i , j \leqslant n + 1 }$, and let $Q _ { n }$ be the matrix of the family $\left( V _ { 0 } , V _ { 1 } , \ldots , V _ { n } \right)$ in the basis $\left( 1 , X , \ldots , X ^ { n } \right)$. Show that $Q _ { n } ^ { \top } G _ { n } Q _ { n } = G _ { n } ^ { \prime }$, where $Q _ { n } ^ { \top }$ is the transpose of the matrix $Q _ { n }$.