A basis $\left( e _ { 1 } , \ldots , e _ { n } \right)$ of $V$ is said to be orthogonal for $q$ if $\widetilde { q } \left( e _ { i } , e _ { j } \right) = 0$ for all $i \neq j$. (a) Show that there exists an orthogonal basis for $q$. Hint: one may consider $\{ x \} ^ { \perp } = \{ y \in V \mid \widetilde { q } ( x , y ) = 0 \}$ and use questions 4c and 6a. (b) Deduce that there exist $a _ { 1 } , \ldots , a _ { n } \in \mathbb { K } - \{ 0 \}$ such that $q \cong \left\langle a _ { 1 } , \ldots , a _ { n } \right\rangle$.
A basis $\left( e _ { 1 } , \ldots , e _ { n } \right)$ of $V$ is said to be orthogonal for $q$ if $\widetilde { q } \left( e _ { i } , e _ { j } \right) = 0$ for all $i \neq j$.
(a) Show that there exists an orthogonal basis for $q$.
Hint: one may consider $\{ x \} ^ { \perp } = \{ y \in V \mid \widetilde { q } ( x , y ) = 0 \}$ and use questions 4c and 6a.
(b) Deduce that there exist $a _ { 1 } , \ldots , a _ { n } \in \mathbb { K } - \{ 0 \}$ such that $q \cong \left\langle a _ { 1 } , \ldots , a _ { n } \right\rangle$.