Proposition 2

a) Let $u$ be an automorphism of V. There exists a unique element of $D^*_{ab}$, denoted by det $u$ and called the determinant of $u$, such that we have

(15) $\omega (u(v_1), . . . , u(v_n)) =$ (det $u$)$\omega (v_1, . . . , v_n)$

for every basis $(v_1, . . . , v_n)$ of V and every $\omega$ in Ω(V).

b) The mapping $u\mapsto$ det $u$ from $\mathbf{G}\mathbf{L}(V)$ to $D^*_{ab}$ is a group homomorphism.

Let $\omega_0$ be an element of Ω(V). The mapping $(v_1, . . . , v_n)\mapsto$ $\omega_0(u(v_1), . . . , u(v_n))$ from B(V) to $D^*_{ab}$ belongs to Ω(V). By Corollary 2 of VIII, p. 451, there exists a unique element $t$ of $D^*_{ab}$ such that we have

for $(v_1, . . . , v_n)$ in B(V). If $\omega$ is another element of Ω(V), then there exists an element $s$ of $D^*_{ab}$ such that

for every basis $(v_1, . . . , v_n)$ of V (loc. cit.). We immediately deduce the relation

This proves a); assertion b) is an immediate consequence of a).