Corollary

Tag 05RQ, permanent. Cite it as [Bourbaki, Tag 05RQ].

For every subset M of E, the rank of M is at most equal to dim E.

If V is the vector subspace of E generated by M, M contains a basis B' of V (no. 1, Theorem 2) and as B' is a free subset of E, it is contained in a basis B of E (no. 1, Theorem 2); then Card(B') $\leq$ Card(B), whence the corollary.

Theorems 2 and 3 immediately imply the following proposition:

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

Cited by

Nothing.

Cites

no. 1, Theorem 2