Proposition 10

The rank of a matrix $X$ over a field $K$ is equal to the rank of its transpose ${}^t X$ over the opposite field $K^0$.

In the notation introduced before Definition 7, the rank of $u$ is equal to that of ${}^t u$ (§ 7, no. 5, Proposition 10) and the proposition therefore follows from no. 4, Proposition 3.

It is thus seen that the rank of $X$ can also be defined as the maximum number of linearly independent rows of $X$ (considering them as elements of the left vector $K$-space $K_s^n$).

The square matrices of order $n$ over a field $K$ correspond bijectively with the endomorphisms of $E = K_d^n$ and form a ring isomorphic to the ring

End$_{\mathbf{K}}$ (E) (no. 7); corresponding to the automorphisms of E are the invertible square matrices.