Affine map

Definition

Let $K$ be a field. Let $A$ be an affine space over a $K$ -vector space $V$ , and let $B$ be an affine space over a $K$ -vector space $W$ . A map $f : A \to B$ is an affine map if there is a linear map $L : V \to W$ such that $f(p + v) = f(p) + L(v) \quad \text{for each point } p \text{ of } A \text{ and each vector } v \text{ of } V.$ The map $L$ is unique. It is the linear part of $f$ , and it satisfies $L(q - p) = f(q) - f(p)$ for all points $p$ and $q$ of $A$ .

Remarks

The affine maps from $\mathbb{R}^n$ to $\mathbb{R}^m$ are the maps $x \mapsto M x + b$ , with $M$ a real $m \times n$ matrix and $b$ a vector of $\mathbb{R}^m$ . An affine map keeps affine combinations. In particular it sends midpoints to midpoints. An affine transformation is a bijective affine map from $A$ to itself. The affine transformations of $A$ form a group under composition. Mathlib's AffineMap is this notion, with the linear part as a field of the structure.

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Affine space concept
Affine map definition

Tag	`0QHX5GGABE`
URI	`m2://0QHX5GGABE`
Kind	definition
Writing	W1 stated
MSC	51N10
Defines	Affine map
Uses	Affine space

Affine map

Definition

Remarks

Reading path

Further reading

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