# mathlibdocumentation

linear_algebra.affine_space.basic

# Affine space #

In this file we introduce the following notation:

• affine_space V P is an alternative notation for add_torsor V P introduced at the end of this file.

We tried to use an abbreviation instead of a notation but this led to hard-to-debug elaboration errors. So, we introduce a localized notation instead. When this notation is enabled with open_locale affine, Lean will use affine_space instead of add_torsor both in input and in the proof state.

Here is an incomplete list of notions related to affine spaces, all of them are defined in other files:

• affine_map: a map between affine spaces that preserves the affine structure;
• affine_equiv: an equivalence between affine spaces that preserves the affine structure;
• affine_subspace: a subset of an affine space closed w.r.t. affine combinations of points;
• affine_combination: an affine combination of points;
• affine_independent: affine independent set of points;
• affine_basis.coord: the barycentric coordinate of a point.

## TODO #

Some key definitions are not yet present.

• Affine frames. An affine frame might perhaps be represented as an affine_equiv to a finsupp (in the general case) or function type (in the finite-dimensional case) that gives the coordinates, with appropriate proofs of existence when k is a field.