Wedge product/Mapping with fixed wedge product/Alternating/Exercise

Let $K$ be a field, and let $V$ denote a $K$ -vector space. Let $u_{1}, \dots, u_{n} \in V$ . Show that for every $k \in ℕ$ , there exists a uniquely determined linear mapping

\overset{k}{⋀} V ⟶ \overset{k + n}{⋀} V

satisfying $v_{1} \land \dots \land v_{k} \mapsto v_{1} \land \dots \land v_{k} \land u_{1} \land \dots \land u_{n}$ .

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