Corollary 2

Tag 0DL6, permanent. Cite it as [Bourbaki, Tag 0DL6].

Algebra, chapter IX, § 5. Propriétés spéciales aux formes bilinéaires alternées

Let A be a commutative field, E a finite-dimensional vector space of dimension n over A. For every bivector $z \in \bigwedge^2 E$ , there exists a basis $(e_i)_{1 \leq i \leq n}$ of E such that

z = e_1 \wedge e_2 + e_3 \wedge e_4 + \cdots + e_{2r-1} \wedge e_{2r} \quad (2r \leq n).

Indeed, it is enough to note that z identifies canonically with an alternating bilinear form on $E^*$ (Chapter III, § 8, No. 2), and to apply Corollary 1 to this form.

By translating Corollary 1 into matrix language, one obtains:

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

Nothing.