I'm working through a proof of the Grassmannian as a subvariety and came across this:
Let V be an n-dimensional vector space, and let $\Lambda$ $\in$ $\wedge$$^k$V be a decomposable multivector of the form $\Lambda$ = v$_1$ $\wedge$ v$_2$ $\wedge$ $\dots$ $\wedge$ v$_k$.
i(v$^*$): $\wedge$$^k$V $\rightarrow$ $\wedge$$^{k-1}$V is a contraction operator defined for v$^*$$\in$ V$^*$ by
$\langle$i(v$^*)$$\Lambda$, $\Xi$$\rangle$ = $\langle$$\Lambda$, v$^*$$\wedge$ $\Xi$$\rangle$ for all $\Xi$ $\in$ ($\wedge$$^{k-1}$V)$^*$$\cong$ $\wedge$$^{k-1}$V$^*$
I'm assuming this last bit is defining the contraction as some sort of inner product, but I have no intuition on how this inner product (of a multivector with a kth-exterior product of the dual space) might work. Can anyone help me out, or at least point me towards something helpful?