Proposition 2

Tag 0644, permanent. Cite it as [Bourbaki, Tag 0644].

Let $M$ be a free $A$ -module with a finite basis $(e_i)_{1 \leq i \leq n}$ and $v$ an endomorphism of $M$ . For every sequence $(x_i)_{1 \leq i \leq n}$ of $n$ elements of $M$ ,

\det(v(x_1), \ldots, v(x_n)) = (\det v) \det(x_1, \ldots, x_n).

If $u$ is the endomorphism of $M$ such that $u(e_i) = x_i$ for all $i$ , then

v(x_i) = (v \circ u)(e_i)

and (5) therefore follows from (2).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

Nothing.