Let $V$ and $W$ be finite dimensional vector spaces, and $f : V \to W$ a homomorphism. Show that $\Lambda^kf:\Lambda^kW^* \to \Lambda^kV^*$, defined by $\Lambda^k\alpha(v_1, ..., v_k)=\alpha(f(v_1), ..., f(v_k))$ gives rise to a well-defined linear map.
I am confused about the definition of the $\Lambda^kf$, it looks that it takes a k-form from $\Lambda^kV^*$ to $\Lambda^kW^*$, so is there a typo with the correction to be: $\Lambda^kf:\Lambda^kV^* \to \Lambda^kW^*$? If not, could you please provide me with the statement that I would need to prove the well-definedness of $\Lambda^kf$?
In the case of the typo, $\Lambda^kf$ is well-defined trivially since both $f$ and $\alpha$ are well-defined by the assumption.
From the Ted Shifrin's comment, the typo was that we do not pass the vectors $v_1, ..., v_k$ all in $V$ to $\alpha \in \Lambda^kW^*$ but we pass these vectors to the form $(\Lambda^kf(\alpha)) \in \Lambda^kV^*$.