Linear function

• 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

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 ${\displaystyle f(x_1,\dots ,x_k)}$ of any finite number of variables, the general formula is

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

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 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.