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In mathematics, the term linear function refers to two distinct but related notions:^[1]

• In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero (a constant polynomial) or one (a linear polynomial).^[2] For distinguishing such a linear function from the other concept, the term affine function is often used.^[3]
• In linear algebra, mathematical analysis,^[4] and functional analysis, a linear function is a kind of function between vector spaces.^[5]

In calculus, analytic geometry and related areas, a linear function is a polynomial of degree one or less, including the zero polynomial. (The latter is a polynomial with no terms, and it is not considered to have degree zero.)

When the function is of only one variable, it is of the form

${\displaystyle f(x)=ax+b,}$

where a and b are constants, often real numbers. The graph of such a function of one variable is a nonvertical line. a is frequently referred to as the slope of the line, and b as the intercept.

If a > 0 then the gradient is positive and the graph slopes upwards.

If a < 0 then the gradient is negative and the graph slopes downwards.

For a function ${\displaystyle f(x_{1},\ldots ,x_{k})}$ of any finite number of variables, the general formula is

${\displaystyle f(x_{1},\ldots ,x_{k})=b+a_{1}x_{1}+\cdots +a_{k}x_{k},}$

and the graph is a hyperplane of dimension k.

A constant function is also considered linear in this context, as it is a polynomial of degree zero or is the zero polynomial. Its graph, when there is only one variable, is a horizontal line.

In this context, a function that is also a linear map (the other meaning of linear functions, see below) may be referred to as a homogeneous linear function or a linear form. In the context of linear algebra, the polynomial functions of degree 0 or 1 are the scalar-valued affine maps.

In linear algebra, a linear function is a map ${\displaystyle f}$ from a vector space ${\displaystyle \mathbf {V} }$ to a vector space ${\displaystyle \mathbf {W} }$ (Both spaces are not necessarily different.) over a same field K such that

${\displaystyle f(\mathbf {x} +\mathbf {y} )=f(\mathbf {x} )+f(\mathbf {y} )}$

${\displaystyle f(a\mathbf {x} )=af(\mathbf {x} ).}$

Here a denotes a constant belonging to the field K of scalars (for example, the real numbers), and x and y are elements of ${\displaystyle \mathbf {V} }$, which might be K itself. Even if the same symbol ${\displaystyle +}$ is used, the operation of addition between x and y (belonging to ${\displaystyle \mathbf {V} }$) is not necessarily same to the operation of addition between ${\displaystyle f\left(\mathbf {x} \right)}$ and ${\displaystyle f\left(\mathbf {y} \right)}$ (belonging to ${\displaystyle \mathbf {W} }$).

In other terms the linear function preserves vector addition and scalar multiplication.

Some authors use "linear function" only for linear maps that take values in the scalar field;^[6] these are more commonly called linear forms.

The "linear functions" of calculus qualify as "linear maps" when (and only when) f(0, ..., 0) = 0, or, equivalently, when the constant b equals zero in the one-degree polynomial above. Geometrically, the graph of the function must pass through the origin.

• Homogeneous function
• Nonlinear system
• Piecewise linear function
• Linear approximation
• Linear interpolation
• Discontinuous linear map
• Linear least squares

• Izrail Moiseevich Gelfand (1961), Lectures on Linear Algebra, Interscience Publishers, Inc., New York. Reprinted by Dover, 1989. ISBN 0-486-66082-6
• Shores, Thomas S. (2007). Applied Linear Algebra and Matrix Analysis. Undergraduate Texts in Mathematics. Springer. ISBN 978-0-387-33195-9.
• Stewart, James (2012). Calculus: Early Transcendentals (7E ed.). Brooks/Cole. ISBN 978-0-538-49790-9.
• Leonid N. Vaserstein (2006), "Linear Programming", in Leslie Hogben, ed., Handbook of Linear Algebra, Discrete Mathematics and Its Applications, Chapman and Hall/CRC, chap. 50. ISBN 1-584-88510-6