Definition 3

Tag 05VU, permanent. Cite it as [Bourbaki, Tag 05VU].

Algebra, chapter II, § 9. Affine spaces and projective spaces

Given two affine spaces $E, E'$ attached to two vector spaces $T, T'$ over the same field $K$ , a mapping $u$ of $E$ into $E'$ is called an affine linear mapping (or an affine mapping) if, for every family $(x_i)_{i \in I}$ of points of $E$ and every family $(\lambda_i)_{i \in I}$ such that $\sum_{i \in I} \lambda_i = 1$ ,

u\left( \sum_{i \in I} \lambda_i x_i \right) = \sum_{i \in I} \lambda_i u(x_i).

(3)

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

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