Proposition 13

*Let E, F be two finite-dimensional vector spaces over a field K. If $u : E \to F$ is a linear mapping of rank r, there exist bases of E and F such that, with respect to these bases,

Every matrix of type $(m, n)$ over K and of rank r is equivalent to a matrix of the form (55).*

The second assertion is trivially equivalent to the first. To show the latter, let $\dim E = n$, $\dim F = m$. The kernel $N = u^{-1}(0)$ is of dimension $n - r$ (§ 7, no. 4, formula (11)); let V be a supplementary subspace of N in E and $(e_i)_{1 \leq i \leq n}$ a basis of E such that $(e_i)_{1 \leq i \leq r}$ is a basis of V and $(e_i)_{r+1 \leq i \leq n}$ a basis of N. Then the $u(e_j)$ $(1 \leq j \leq r)$ form a basis of $u(E)$; hence there exists a basis $(f_j)_{1 \leq j \leq m}$ of F such that $f_j = u(e_j)$ for $1 \leq j \leq r$ (§ 7, no. 1, Theorem 2) and clearly with respect to the bases $(e_i)$ and $(f_j)$ the matrix $M(u)$ is given by (55).