Definition 3

Tag 05S9, permanent. Cite it as [Bourbaki, Tag 05S9].

*Let $E, F$ be two vector spaces over a field $K$ . For every linear mapping $u$ of $E$ into $F$ , the dimension of the subspace $u(E)$ of $F$ is called the rank of $u$ and denoted by $\mathrm{rg}(u)$ .

If $N = \mathrm{Ker}(u)$ , $E/N$ is isomorphic to $u(E)$ , whence the relation

\mathrm{rg}(u) = \mathrm{codim}_E(\mathrm{Ker}(u))

and therefore

\mathrm{rg}(u) + \dim(\mathrm{Ker}(u)) = \dim E.

Moreover, by formula (3)

\mathrm{rg}(u) + \dim(\mathrm{Coker}(u)) = \dim F.

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

Cited by

Appendix 4. Trace of an Endomorphism of Finite Rank

Cites

Nothing.