Bourbaki

Exercise 12, § 7

Algebra, II, 403

Let E, F be two vector spaces and u,vu, v two linear mappings of E into F. (a) Show that the following inequalities hold:

rg(v)≤rg(u+v)+rg(u)rg(u+v)≤inf⁡(dim⁡(E),dim⁡(F),rg(u)+rg(v)). \begin{align*} \mathrm{rg}(v) &\leq \mathrm{rg}(u + v) + \mathrm{rg}(u) \\ \mathrm{rg}(u + v) &\leq \inf(\dim(E), \dim(F), \mathrm{rg}(u) + \mathrm{rg}(v)). \end{align*}
If E and F are finite-dimensional, show that rg(u+v)\mathrm{rg}(u + v) can take any integer value satisfying the inequalities
∣rg(u)−rg(v)∣≤rg(u+v)≤inf⁡(dim⁡(E),dim⁡(F),rg(u)+rg(v)). |\mathrm{rg}(u) - \mathrm{rg}(v)| \leq \mathrm{rg}(u + v) \leq \inf(\dim(E), \dim(F), \mathrm{rg}(u) + \mathrm{rg}(v)).
(b) Show that
dim⁡(Ker(u+v))≤dim⁡(Ker(u)∩Ker(v))+dim⁡(Im(u)∩Im(v)). \dim(\mathrm{Ker}(u + v)) \leq \dim(\mathrm{Ker}(u) \cap \mathrm{Ker}(v)) + \dim(\mathrm{Im}(u) \cap \mathrm{Im}(v)).
(c) Show that
dim⁡(Coker(u+v))≤dim⁡(Coker(u))+rg(v). \dim(\mathrm{Coker}(u + v)) \leq \dim(\mathrm{Coker}(u)) + \mathrm{rg}(v).
(If V is a supplementary subspace of Ker(v)\mathrm{Ker}(v) in E, note that
u(Ker(v))⊂Im(u+v)anddim⁡(u(V))≤rg(v). u(\mathrm{Ker}(v)) \subset \mathrm{Im}(u + v) \quad \text{and} \quad \dim(u(V)) \leq \mathrm{rg}(v).
)

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