Equivalent Properties of Nondegenerate Symmetric Covariant 2-Tensor

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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.


Proof




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