We return to the case where $\mathbb { K }$ is an arbitrary field of characteristic zero. Let $V$ be a $\mathbb { K }$-vector space of finite dimension and $q \in \mathcal { Q } ( V )$. Show that there exists a unique non-negative integer $m$ and an anisotropic quadratic form $q _ { \text {an} }$, unique up to isometry, such that $q \cong q _ { an } \perp m \cdot h$ where $m \cdot h = h \perp \cdots \perp h$ is the orthogonal sum of $m$ copies of $h$ and $h$ is the quadratic form defined by $h \left( x _ { 1 } , x _ { 2 } \right) = x _ { 1 } x _ { 2 }$ (introduced in question 6b). Hint: one may use question 6b and the previous question.
We return to the case where $\mathbb { K }$ is an arbitrary field of characteristic zero. Let $V$ be a $\mathbb { K }$-vector space of finite dimension and $q \in \mathcal { Q } ( V )$. Show that there exists a unique non-negative integer $m$ and an anisotropic quadratic form $q _ { \text {an} }$, unique up to isometry, such that $q \cong q _ { an } \perp m \cdot h$ where $m \cdot h = h \perp \cdots \perp h$ is the orthogonal sum of $m$ copies of $h$ and $h$ is the quadratic form defined by $h \left( x _ { 1 } , x _ { 2 } \right) = x _ { 1 } x _ { 2 }$ (introduced in question 6b).
Hint: one may use question 6b and the previous question.