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 .