**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**: A covariant $k$-tensor is called

*alternating* if its value changes sign by interchanging any pair of its arguments:

\[ T(X_{1}, \dots, X_{i}, \dots, X_{j}, \dots, X_{k}) = - T(X_{1}, \dots, X_{j}, \dots, X_{i}, \dots, X_{k}) \]whenever $ 1 \leq i < j \leq k $.

Show that the following are equivalent for a covariant $k$-tensor $T$:

- $T$ is alternating.
- For any vectors $X_{1}, \dots, X_{k} \in V $ and any permutation $ \sigma \in S_{k} $, it holds that

\[ T( X_{\sigma(1)}, \dots, X_{\sigma(k)}) = \text{sgn}(\sigma) T(X_{1}, \dots, X_{k}) \] - $T$ gives zero whenever two of its arguments are equal, that is

\[ T(X_{1}, \dots, Y, \dots, Y, \dots X_{k}) = 0 \] - $ T(X_{1}, \dots, X_{k}) = 0 $ whenever the vectors $ X_{1}, \dots, X_{k} \in V $ are linearly dependent.
- The components $T_{i_{1} \dots i_{k}} = T(E_{i_{1}}, \dots, E_{i_{k}})$ of $T$ with respect to any basis $ \left\{ E_{i} \right\} $ of $V$ change sign whenever two indices are interchanged.