Suppose we have $A: V_{1}\to V_{2}$ where $V_{1},V_{2}$ are real vector spaces. Then $A^{\star}:\mathcal{J}^{k}(V_{2})\to \mathcal{J}^{k}(V_{1})$ where $\mathcal{J}(V):=\{\text{space of all $k$-tensors on $V$ }\}$
How to show by using definitions that
$A^{\star}:\Lambda^{k}(V_{2})\to \Lambda^{k}(V_{1})$
where $\Lambda^{k}(V):=\{\text{space of alternating $k$-tensors ($k$-forms on $V$)}\}$
Clearly if $\omega\in \Lambda^{k}(V)$, since $ \Lambda^{k}(V)\subset\mathcal{J}(V)$ then also $\omega\in \mathcal{J}(V)$, apart from that I have no idea what should really be shown here (it seems natural that it is like that...)but I wonder how this can be showed rigorously based on some definitions.
Thank you in advance.