We say that $q \in \mathcal{Q}(V)$ is isotropic if there exists $x \in V - \{ 0 \}$ such that $q ( x ) = 0$. Otherwise, we say that $q$ is anisotropic. (a) Prove that there exists $x \in V$ such that $q ( x ) \neq 0$. (b) We denote by $h$ the quadratic form on $\mathbb { K } ^ { 2 }$ defined by $h \left( x _ { 1 } , x _ { 2 } \right) = x _ { 1 } x _ { 2 }$ (we do not ask you to verify that $h$ is a quadratic form). Show that if $V$ is of dimension two and $q$ is isotropic then $q$ is isometric to $h$. (c) Prove that if $q \in \mathcal { Q } ( V )$ is isotropic, then $q : V \rightarrow \mathbb { K }$ is surjective.
We say that $q \in \mathcal{Q}(V)$ is isotropic if there exists $x \in V - \{ 0 \}$ such that $q ( x ) = 0$. Otherwise, we say that $q$ is anisotropic.
(a) Prove that there exists $x \in V$ such that $q ( x ) \neq 0$.
(b) We denote by $h$ the quadratic form on $\mathbb { K } ^ { 2 }$ defined by $h \left( x _ { 1 } , x _ { 2 } \right) = x _ { 1 } x _ { 2 }$ (we do not ask you to verify that $h$ is a quadratic form). Show that if $V$ is of dimension two and $q$ is isotropic then $q$ is isometric to $h$.
(c) Prove that if $q \in \mathcal { Q } ( V )$ is isotropic, then $q : V \rightarrow \mathbb { K }$ is surjective.