Exercise 18, § 7

Let E, F be two vector spaces over a field K and $u, v$ two linear mappings of E into F. For there to exist an automorphism $f$ of E and an automorphism $g$ of F such that $v = g \circ u \circ f$, it is necessary and sufficient that $\operatorname{rg}(u) = \operatorname{rg}(v)$, $\dim(\operatorname{Ker}(u)) = \dim(\operatorname{Ker}(v))$ and $\dim(\operatorname{Coker}(u)) = \dim(\operatorname{Coker}(v))$.