| # Linear and Affine Maps Here are some equivalent definitions of linear or affine functions and maps. A map is linear if and only if either one of the following conditions hold. preserves scaling and addition of its arguments: for every , and , ; and for every , . vanishes at the origin: , and transforms any line segment in into another segment in : is differentiable, vanishes at the origin, and the matrix of its derivatives is constant. There exist such that A map is affine if and only if the map with values is linear. Example: the function , with values is linear when , and affine otherwise. In contrast, the function with values is not linear nor affine, whatever the value of .