Linear function

{wiki}

= Linear
{synonym}

= Linearly
{synonym}

The term is not very clear, as it could either mean:
* a <real number> function whose graph is a line, i.e.:
  $$
  f(x) = ax + b
  $$
  or for higher dimensions, a <hyperplane>:
  $$
  f(x_1, x_2, \ldots, x_n) = c_1 x_1 + c_2 x_2 + \ldots + c_n x_n + b
  $$
* a <linear map>. Note that the above linear functions are not linear maps unless $b = 0$ (known as the homogeneous case), because e.g.:
  $$
  f(x + y) = ax + ay + b
  $$
  but
  $$
  f(x) + f(y) = ax + b + ay + b
  $$
  For this reason, it is better never to refer to linear maps as linear functions.
