Exercise 13, § 7

Let E, F, G be three vector spaces and $u : E \to F, v : F \to G$ two linear mappings. (a) Show that there exists a decomposition of E as a direct sum of $\mathrm{Ker}(u)$ and two subspaces M, N such that $\mathrm{Ker}(v \circ u) = M \oplus \mathrm{Ker}(u)$ and

(b) $\dim(\mathrm{Ker}(u)) \leq \dim(\mathrm{Ker}(v \circ u)) \leq \dim(\mathrm{Ker}(u)) + \dim(\mathrm{Ker}(v));$ if $\mathrm{Ker}(u)$ and $\mathrm{Ker}(v)$ are finite-dimensional, show that $\dim(\mathrm{Ker}(v \circ u))$ can take every integer value satisfying the above inequalities. (c) The following equalities hold: If F is finite-dimensional, show that $\mathrm{rg}(v \circ u)$ can take any integer value satisfying the inequality