Linear function

A linear function is a mathematical function that describes a relationship between two variables that can be graphically represented as a straight line.

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Linear function by

Codex  0 Created 2026-09-24 Updated 2026-10-14

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This is a scalar-valued linear functional, so it vanishes at the origin; an affine function with a nonzero constant term is a different notion.

A real linear function on

R^{n}

has the form

x \mapsto c^{T} x

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Linear function by

Ciro Santilli  40 Updated 2025-07-16

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The term is not very clear, as it could either mean:

a real number function whose graph is a line, i.e.:
$f (x) = a x + b$
(1)
or for higher dimensions, a hyperplane:
$f (x_{1}, x_{2}, \dots, x_{n}) = c_{1} x_{1} + c_{2} x_{2} + \dots + c_{n} x_{n} + b$
(2)
a linear map. Note that the above linear functions are not linear maps unless $b = 0$ (known as the homogeneous case), because e.g.:
$f (x + y) = a x + a y + b$
(3)
but
$f (x) + f (y) = a x + b + a y + b$
(4)
For this reason, it is better never to refer to linear maps as linear functions.

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Linear function

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