In the last lecture we started speaking about hodge star operator. Let $E$ be a $n$ dimensional vector space with a non degenerate bilinear form $g$.
$\mathcal{O}$ the orientation line of $E$, i.e. the space of signed densities on $E$. By density I mean a map $$ \rho \colon \Lambda^nE^* \to \mathbb{R}$$ such that $\rho(\lambda w)=\text{sign}(\lambda)\rho(w)$ for every $\lambda$ scalar and $w \in \Lambda^nE^*$.
Now define the hodge star operator in this way
$$* \colon \Lambda^p E \to \Lambda^{n-p}E \otimes \mathcal{O} $$ $$ \eta \to * \eta$$ where $*\eta$ is the unique element such that $$(w \wedge * \eta)= (\Lambda^p g)(w,\eta) \text{vol}_g$$ for every $w \in \Lambda^p E$ and $\text{vol}_g$ is the volume element associate to an orthonormal basis $e_i$ of $E$ and the bilinear form $g$.
Now define $int_x$ is the interior product which associate to a covector in $\Lambda^kE^*$ a covector in $\Lambda^{k-1}E^*$ jus "filling the first entry of it with $x$". Define now $x^b \colon E \to E^*$ to be map which associate to $x$ the linear map $v \mapsto g(v,x)$.
Now I'm asked to prove the followed "equality" (I'm supposed to find the right sign of it).
Prove that, for $\eta \in \Lambda E^* \otimes \mathcal{O}$ $$ int_x \eta = \pm *( x^b \wedge * \eta)$$
Obviously I don't know where to start, but I'm not here to ask for a solution (yet), I'm here to trying understanding how can I interpret the equality above. This is my first given exercise with the hodge operator so it should be kind of easy...
The definition seen during lectures of $*$ doesn't involve covectors, and here I have covectors. Moreover, even for the interior product I've problem with the domain, because $\eta$ is tensored with the orientation line. So is there a way to "adapt" my definitions to give this writing a meaning?
Thanks in advance, and I apologise in advance if the question is not clear enough or the definitions are not standard, but every books use its own definitions.