Corollary 2

If $M(x)$ and $M'(x)$ are the matrices with one column of the same element $x \in E$ with respect to the bases $(e_i)$ and $(e'_i)$ respectively, then

This is a special case of Proposition 6, applied to the mapping $\theta_x : a \mapsto xa$ of $A_d$ to $E$ (no. 4, Remark 1).

Formula (35) is equivalent to

for the elements $x_i$ and $x'_i$ of the matrices $M(x)$ and $M'(x)$ respectively. Formulae (36 D) are called formulae of change of coordinates. Observe that they express the components of $x$ relative to the "old" basis $(e_i)$ as functions of the components of $x$ relative to the "new" basis $(e'_i)$ and the elements of $P$, that is the components of the "new" basis relative to the "old" basis.