Proposition 5

Let $(e_i), (e'_i)$ be two bases of n elements of E and P the matrix of passage from $(e_i)$ to $(e'_i)$. If $(e_i^*)$ and $({e'_i}^*)$ are the respective dual bases of $(e_i)$ and $(e'_i)$, the matrix of passage from $(e_i^*)$ to $({e'_i}^*)$ is the contragredient $^tP^{-1}$ of P.

The transpose of the identity mapping $l_E$ is the identity mapping $l_{E^*}$; by Proposition 3, no. 4, the matrix of $l_{E^*}$ with respect to the bases $({e'_i}^*)$ and $(e_i^*)$ (in that order) is the transpose of the matrix of $l_E$ with respect to the bases $(e_i)$ and $(e'_i)$ (in that order), that is the transpose of $P^{-1}$.