Step in a proof about alternating operators

Question

The theorem is that if $f$ is a $k$-linear function on a vector space $V$, then the $k$-linear function $Af$ is alternating.

$\def\sgn{\operatorname{sgn}}Af=\sum (\sgn \sigma)\sigma f$

Proof: $$\omega(Af)=\sum_{\sigma}(\sgn\sigma)\omega(\sigma f)\implies\sum_{\sigma}(\sgn\sigma)(\omega\sigma)f=(\sgn\omega)\sum_{\sigma}(\sgn\omega\sigma)(\omega\sigma)f$$

The last step here I do not understand. The next step after that will be $$(\sgn\omega)Af$$

And it will complete the proof. Can someone please show why the previous step is the case?

Thanks in advance!

P.S. I am using Tu's text "introduction to manifolds"

Answer 1

Since $\sigma = \omega^{-1} \omega \sigma$, it follows that

$$\textrm{sgn} \:\sigma = \textrm{sgn} \left(\omega^{-1} \omega\sigma\right)=\textrm{sgn} (\omega^{-1}) \:\textrm{sgn} (\omega \sigma) = \textrm{sgn} (\omega)\: \textrm{sgn} (\omega \sigma).$$