Exercise 17, § 11

Let K be a commutative field and E a vector space over K.

(a) Let $z$ be a $p$-vector $\neq 0$ over E and let $V_z$ be the vector subspace of E consisting of the vectors $x$ such that $z \wedge x = 0$. Then $\dim(V_z) \leq p$; if $(x_i)_{1 \leq i \leq q}$ is a free system of vectors of $V_z$, there exists a $(p - q)$-vector $v$ such that $z = v \wedge x_1 \wedge \cdots \wedge x_q$. If E is finite-dimensional, then $V_z = \overline{\phi}(M_z^0)$, where $M_z^0$ is the orthogonal in $E^*$ of the subspace $M_z$ associated with $z$. For $z$ to be a pure $p$-vector, it is necessary and sufficient that $\dim(V_z) = p$ and then $V_z = M_z$.

(b) Let $v$ be a pure $p$-vector, $w$ a pure $q$-vector and V and W the subspaces of E associated respectively with $v$ and $w$. In order that $V \subset W$, it is necessary and sufficient that $q \geq p$ and that there exist a $(q - p)$-vector $u$ such that $w = u \wedge v$. In order that $V \cap W = \{0\}$, it is necessary and sufficient that v ∧ w ≠ 0; the subspace V + W is then associated with the pure (p + q)-vector v ∧ w.

(c) Let u and v be two pure p-vectors and U and V the subspaces associated respectively with u and v. For u + v to be pure, it is necessary and sufficient that dim(U ∩ V) ≥ p − 1.