Equivalent Properties of Nondegenerate Symmetric Covariant 2-Tensor

Theorem
Let $V$ and $V^*$ be a finite dimensional vector space and its dual.

Let $q$ be a symmetric covariant 2-tensor on $V$.

Let $\tuple {\epsilon^i}$ be any basis of $V^*$

Then the following properties are equivalent:


 * $q$ is nondegenerate.


 * $\forall v \in V : v \ne 0 : \exists w \in V : \map q {v, w} \ne 0$


 * If $q = q_{ij} \epsilon^i \epsilon^j$, then the matrix $\paren {q_{ij}}$ is invertible.