\[ \newcommand{\d}{\mathrm{d}} \newcommand{\e}{\mathrm{e}} \newcommand{\i}{\mathrm{i}} \]

A covariant \(k\)-tensor is said to be alternating (or antisymmetric or skew-symmetric) if it change sign when two arguments are interchanged.

The subspace of all alternating \(k\)-tensors on \(V\) is denoted by \(\Lambda^k(V^*)\subset T^k(V^*)\)

Remarks
  • Consider an alternating tensor \(\alpha\). Then applying a linear map \(T\) on every argument, is the same as multiplying \(\alpha\) with \(\det T\) (see (0x66d1b291) ), i.e. \begin{equation*} \alpha(Tv_1,\ldots,Tv_k)=(\det T)\alpha(v_1,\ldots, v_k). \end{equation*}

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