Linear function

In mathematics, the term linear function refers to two distinct but related notions:

• 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). For distinguishing such a linear function from the other concept, the term affine function is often used.
• In linear algebra, mathematical analysis, and functional analysis, a linear function is a kind of function between vector spaces.

As a polynomial function

File:Linear Function Graph.svg

Graphs of two linear functions.

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

$f(x)=ax+b,$

where Template:Mvar and Template:Mvar are constants, often real numbers. The graph of such a function of one variable is a nonvertical line. Template:Mvar is frequently referred to as the slope of the line, and Template:Mvar 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 $f(x_1,\dots ,x_k)$ of any finite number of variables, the general formula is

$f(x_1,\dots ,x_k)=b+a_1{x}_1+\cdots +a_k{x}_k,$

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 the 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.

As a linear map

File:Integral as region under curve.svg

An integral of an integrable function is a linear map from a vector space of integrable functions to real numbers (that is also a vector space).

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

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

$f(a\mathbf{𝐱})=af(\mathbf{𝐱}).$

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

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; 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 Template:Mvar equals zero in the one-degree polynomial above. Geometrically, the graph of the function must pass through the origin.

See also

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

References

• Izrail Moiseevich Gelfand (1961), Lectures on Linear Algebra, Interscience Publishers, Inc., New York. Reprinted by Dover, 1989. Template:Isbn
• 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. Template:Isbn