Exercise 16, § 7

Let E, F be two vector spaces over K and $u : E \to F$ a linear mapping. $u$ is said to be of finite index if $\operatorname{Ker}(u)$ and $\operatorname{Coker}(u)$ are finite-dimensional and the number $d(u) = \dim(\operatorname{Ker}(u)) - \dim(\operatorname{Coker}(u))$ is then called the index of $u$.

Let G be a third vector space over K and $v : F \to G$ a linear mapping. Show that if two of the three linear mappings $u, v, v \circ u$ are of finite index, so is the third and $d(v \circ u) = d(u) + d(v)$ (use Exercise 13 (a)).