Multilinear mapping/Constant factor/Alternating/Linear product mapping first/Exercise

Let ${}K$ be a field, let ${}V$ and ${}W$ be ${}K$-vector spaces, and let

$\varphi \colon V\longrightarrow W$

denote a ${}K$-linear mapping, Let

$\triangle \colon W^{m}\longrightarrow K$

denote a multilinear mapping. Show that the composed mapping

$V^{m}\longrightarrow K,(v_{1},\ldots ,v_{m})\longmapsto \triangle \left(\varphi (v_{1}),\,\ldots ,\,\varphi (v_{m})\right),$

is multilinear. Moreover, show that, if ${}\triangle$ is alternating, then also ${}\triangle \circ \varphi ^{n}$ is alternating, and that, if ${}\varphi$ is bijective, also the converse holds.

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