In mathematics, the term linear function refers to two distinct but related notions:[1]
See main article: article and Linear function (calculus).
In calculus, analytic geometry and related areas, a linear function is a polynomial of degree one or less, including the zero polynomial (the latter not being considered to have degree zero).
When the function is of only one variable, it is of the form
f(x)=ax+b,
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(x1,\ldots,xk)
f(x1,\ldots,xk)=b+a1x1+ … +akxk,
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) 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.
See main article: article and Linear map.
In linear algebra, a linear function is a map f between two vector spaces such that
f(x+y)=f(x)+f(y)
f(ax)=af(x).
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), or, equivalently, when the constant equals zero in the one-degree polynomial above. Geometrically, the graph of the function must pass through the origin.