Bourbaki

Corollary 3

Tag 05S0, permanent. Cite it as [Bourbaki, Tag 05S0].

Algebra, chapter II, § 7. Vector spaces

If M and N are two subspaces of a vector space E, then

(7)

dim⁡(M+N)+dim⁡(M∩N)=dim⁡M+dim⁡N. \dim(M + N) + \dim(M \cap N) = \dim M + \dim N.

It suffices to apply Corollary 2 to the exact sequence

0→M∩N→M⊕N→M+N→0 0 \to M \cap N \to M \oplus N \to M + N \to 0

(§ 1, no. 8, Proposition 10) taking account of the fact that

dim⁡(M⊕N)=dim⁡M+dim⁡N \dim(M \oplus N) = \dim M + \dim N

(no. 2, Proposition 2).

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

Cited by

Cites

Transcribed from the printed edition, 1998, Springer. Not translated.