The definition of the contraction $\iota_\wedge (e)$

Question

Let $V$ be vector space and let $\wedge V$ be the exterior algebra of $V$.

If $e \in V$, what is the definition of the operator which is called the contraction by $e$ : $$\iota_\wedge(e): \wedge V \rightarrow \wedge V$$

Answer 1

First define $\iota_\wedge(e):\Lambda^p(V)\to\Lambda^{p-1}(V)$ by $$ \iota_\wedge(e)\omega(x_1,\ldots,x_{p-1}) = \omega(e,x_1,\ldots,x_{p-1}). $$ Then extend this to $\Lambda(V)$ by linearity.