semiaffine transformation of is a bijection from onto itself satisfying: For every -dimensional affine subspace of , then is also a -dimensional affine subspace of . If and are parallel affine subspaces of , then and are parallel. These two conditions are satisfied by affine transformations, and express what is precisely meant by the expression that " preserves parallelism". These conditions are not independent as the second follows from the first. Furthermore, if the field has at least three elements, the first condition can be simplified to: is a collineation, that is, it maps lines to lines. Structure By the definition of an affine space, acts on , so that, for every pair in there is associated a point in . We can denote this action by . Here we use the convention that are two interchangeable notations for an element of . By fixing a point in one can define a function by . For any , this function is one-to-one, and so, has an inverse function given by . These functions can be used to turn into a vector space (with respect to the point ) by defining: and This vector space has origin and formally needs to be distinguished from the affine space , but common practice is to denote it by the same symbol and mention that it is a vector space after an origin has been specified. This identification permits points to be viewed as vectors and vice versa. For any linear transformation of , we can define the function by Then is an affine transformation of which leaves the point fixed. It is a linear transformation of , viewed as a vector space with origin . Let be any affine transformation of . Pick a point in and consider the translation of by the vector , denoted by . Translations are affine transformations and the composition of affine transformations is an affine transformation. For this choice of , there exists a unique linear transformation of such that That is, an arbitrary affine transformation of is the composition of a linear transformation of (viewed as a vector space) and a translation of . This representation of affine transformations is often taken as the definition of an affine transformation (with the choice of origin being implicit). Representation As shown above, an affine map is the composition of two functions: a translation and a linear map. Ordinary