Let $\mathcal { B } : = \left( e _ { 1 } , \ldots , e _ { n } \right)$ be a basis of $V$. We associate to every symmetric bilinear form $b$ on $V$ a symmetric matrix $\Phi _ { \mathcal { B } } ( b ) : = \left( b \left( e _ { i } , e _ { j } \right) \right) _ { i , j = 1 \ldots n }$ called the matrix of $b$ in the basis $\mathcal { B }$. We recall that $b \mapsto \Phi _ { \mathcal { B } } ( b )$ is an isomorphism between the vector space of symmetric bilinear forms on $V$ and that of square symmetric matrices of size $n$.
(a) Prove that a quadratic form $q$ on $V$ is non-degenerate if and only if the determinant $\operatorname { det } \left( \Phi _ { \mathcal { B } } ( \tilde { q } ) \right)$ is non-zero.
(b) What is the matrix of $\left\langle a _ { 1 } , \ldots , a _ { n } \right\rangle$ in the canonical basis of $\mathbb { K } ^ { n }$ ? Deduce that $\left\langle a _ { 1 } , \ldots , a _ { n } \right\rangle \in \mathcal { Q } \left( \mathbb { K } ^ { n } \right)$.