**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 $\omega$:

- $\omega$ is non-degenerate.
- The matrix $(\omega_{ij})$ representing $\omega$ in terms of any basis is invertible.
- The linear map $ \tilde{\omega} \ \colon V \to V^{*} \ , \ \tilde{\omega}(X)(Y) = \omega(X,Y) $ is non-singular.