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 Post subject: Tensors - Part 3Posted: Sun Dec 25, 2016 6:34 pm
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Definition 1: Let $V$ be a finite-dimensional $\mathbb{R}$-vector space and let $k$ be a non-negative integer. A covariant $k$-tensor is a multilinear function $T \ \colon V \times \dots \times V \to \mathbb{R}$.

Definition 2: Let $V$ be a finite-dimensional $\mathbb{R}$-vector space. A 2-tensor $\omega$ on $V$ is called non-degenerate if the following implication holds:
$\omega(X,Y) = 0 \ , \forall Y \in V \implies X = 0$

Show that the following are equivalent for a $2$-tensor $\omega$:
1. $\omega$ is non-degenerate.
2. The matrix $(\omega_{ij})$ representing $\omega$ in terms of any basis is invertible.
3. The linear map $\tilde{\omega} \ \colon V \to V^{*} \ , \ \tilde{\omega}(X)(Y) = \omega(X,Y)$ is non-singular.

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