grandes-ecoles 2014 Q5

grandes-ecoles · France · x-ens-maths1__mp Groups Symplectic and Orthogonal Group Properties

Let $q \in \mathcal { Q } ( V )$ and $q ^ { \prime } \in \mathcal { Q } \left( V ^ { \prime } \right)$ where $V ^ { \prime }$ is a $\mathbb { K }$-vector space of finite dimension. Prove that $O ( q )$ is a subgroup of $\mathrm { GL } ( V )$ and that if $q \cong q ^ { \prime }$, then $O ( q )$ and $O \left( q ^ { \prime } \right)$ are two isomorphic groups.

Let $q \in \mathcal { Q } ( V )$ and $q ^ { \prime } \in \mathcal { Q } \left( V ^ { \prime } \right)$ where $V ^ { \prime }$ is a $\mathbb { K }$-vector space of finite dimension. Prove that $O ( q )$ is a subgroup of $\mathrm { GL } ( V )$ and that if $q \cong q ^ { \prime }$, then $O ( q )$ and $O \left( q ^ { \prime } \right)$ are two isomorphic groups.

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