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 Post subject: Tensors - Part 2Posted: Sun Dec 25, 2016 6:25 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: 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 $T$:
1. $T$ is alternating.
2. For any vectors $X_{1}, \dots, X_{k} \in V$ and any permutation $\sigma \in S_{k}$, it holds that
$T( X_{\sigma(1)}, \dots, X_{\sigma(k)}) = \text{sgn}(\sigma) T(X_{1}, \dots, X_{k})$
3. $T$ gives zero whenever two of its arguments are equal, that is
$T(X_{1}, \dots, Y, \dots, Y, \dots X_{k}) = 0$
4. $T(X_{1}, \dots, X_{k}) = 0$ whenever the vectors $X_{1}, \dots, X_{k} \in V$ are linearly dependent.
5. 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.

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