Linear function


In mathematics, the term linear function refers to two distinct but related notions:
In calculus, analytic geometry and related areas, a linear function is a polynomial of degree one or less, including the zero polynomial.
When the function is of only one variable, it is of the form
where and are constants, often real numbers. The graph of such a function of one variable is a nonvertical line. is frequently referred to as the slope of the line, and 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 of any finite number of variables, the general formula is
and the graph is a hyperplane of dimension.
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 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

In linear algebra, a linear function is a map from a vector space to a vector space over a same field such that
Here denotes a constant belonging to the field of scalars, and and are elements of, which might be itself. Even if the same symbol is used, the operation of addition between and is not necessarily same to the operation of addition between and .
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 , or, equivalently, when the constant equals zero in the one-degree polynomial above. Geometrically, the graph of the function must pass through the origin.