Lemma 1

— Let $U$ and $V$ be two vector spaces of finite dimension over a field $L$. Every linear mapping $f : U \to V$ possesses an index equal to $[V : L] - [U : L]$.

Let $N$ be the kernel, $I$ the image and $C$ the cokernel of $f$. We have $C = V/I$ and $I$ is isomorphic to $U/N$; thus we have $[U : L] = [N : L] + [I : L]$ and $[V : L] = [C : L] + [I : L]$, whence the result.

$^1$ Not necessarily commutative. $^2$ On the left.