Exercise 20, § 11

Let E be an n-dimensional vector space, z a pure p-vector over E, V the vector subspace of E associated with z, $u^*$ a pure q-vector over $E^*$, W' the subspace of $E^*$ associated with $u^*$ and W the subspace of E, of dimension $n - q$, orthogonal to W'. If $q < p$, $u^* \lrcorner z$ is a pure $(p - q)$-vector over E, which is zero if $\dim(V \cap W) > p - q$; if $\dim(V \cap W) = p - q$, $V \cap W$ is associated with $u^* \lrcorner z$.