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Let X be an affine space over a field k, and V be its associated vector space. An affine transformation is a bijection f from X onto itself that is an affine map; this means that a linear map g from V to V is well defined by the equation () = (); here, as usual, the subtraction of two points denotes the free vector from the second point to the first one, and "well-defined" means that ...
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication.
Another important family of examples are the linear maps centred at an origin: given a point and a linear map , one may define an affine map ,: by , = + for every in . After making a choice of origin b {\displaystyle b} , any affine map may be written uniquely as a combination of a translation and a linear map centred at b {\displaystyle b} .
A constant function is also considered linear in this context, as it is a polynomial of degree zero or is the zero polynomial. Its graph, when there is only one variable, is a horizontal line. In this context, a function that is also a linear map (the other meaning) may be referred to as a homogeneous linear function or a linear form.
In projective geometry, affine space means the complement of a hyperplane at infinity in a projective space. Affine space can also be viewed as a vector space whose operations are limited to those linear combinations whose coefficients sum to one, for example 2x − y, x − y + z, (x + y + z)/3, ix + (1 − i)y, etc.
The subgroup of the special affine group consisting of those transformations whose linear part has determinant 1 is the group of orientation- and volume-preserving maps. Algebraically, this group is a semidirect product S L ( V ) ⋉ V {\displaystyle SL(V)\ltimes V} of the special linear group of V {\displaystyle V} with the translations.
Since the graph of an affine(*) function is a line, the graph of a piecewise linear function consists of line segments and rays. The x values (in the above example −3, 0, and 3) where the slope changes are typically called breakpoints, changepoints, threshold values or knots.
Roughly, affine spaces are vector spaces whose origins are not specified. [96] More precisely, an affine space is a set with a free transitive vector space action. In particular, a vector space is an affine space over itself, by the map , (,) +.