Definition:Nondegenerate Symmetric Covariant 2-Tensor

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

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

Let $\hat q : V \to V^*$ be a linear mapping such that:

$\forall v, w \in V : \map {\map {\hat q} v} w := \map q {v, w}$

Suppose $\hat q$ is an isomorphism.

Then $q$ is said to be nondegenerate.