Let $q \in \mathcal { Q } ( V )$ be a non-degenerate quadratic form on $V$. (a) Let $V ^ { \prime }$ be a $\mathbb { K }$-vector space of finite dimension and $q ^ { \prime }$ a quadratic form on $V ^ { \prime }$. Prove that if $q$ and $q ^ { \prime }$ are isometric, then $q ^ { \prime }$ is in $\mathcal { Q } \left( V ^ { \prime } \right)$, that is, non-degenerate. (b) For $x \neq 0$, we denote $\{ x \} ^ { \perp } : = \{ y \in V \mid \widetilde { q } ( x , y ) = 0 \}$. Show that $\{ x \} ^ { \perp }$ is a vector subspace of $V$ of dimension $n - 1$. (c) Under what condition on $x$ is the subspace $\{ x \} ^ { \perp }$ a complement of the line $\mathbb { K } x$ in $V$ ?
Let $q \in \mathcal { Q } ( V )$ be a non-degenerate quadratic form on $V$.
(a) Let $V ^ { \prime }$ be a $\mathbb { K }$-vector space of finite dimension and $q ^ { \prime }$ a quadratic form on $V ^ { \prime }$. Prove that if $q$ and $q ^ { \prime }$ are isometric, then $q ^ { \prime }$ is in $\mathcal { Q } \left( V ^ { \prime } \right)$, that is, non-degenerate.
(b) For $x \neq 0$, we denote $\{ x \} ^ { \perp } : = \{ y \in V \mid \widetilde { q } ( x , y ) = 0 \}$. Show that $\{ x \} ^ { \perp }$ is a vector subspace of $V$ of dimension $n - 1$.
(c) Under what condition on $x$ is the subspace $\{ x \} ^ { \perp }$ a complement of the line $\mathbb { K } x$ in $V$ ?