Prove that $\left\langle a _ { 1 } , \ldots , a _ { n } \right\rangle$ is indeed a quadratic form on $\mathbb { K } ^ { n }$, where $\left\langle a _ { 1 } , \ldots , a _ { n } \right\rangle$ denotes the quadratic form $q$ defined on $\mathbb { K } ^ { n }$ by the formula
$$q \left( x _ { 1 } , \ldots , x _ { n } \right) = a _ { 1 } x _ { 1 } ^ { 2 } + \cdots + a _ { n } x _ { n } ^ { 2 }$$