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 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,
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 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.
See main article: article and Linear map.
In linear algebra, a linear function is a map
f
V
W
f(x+y)=f(x)+f(y)
f(ax)=af(x).
V
+
V
f\left(x\right)
f\left(y\right)
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), or, equivalently, when the constant equals zero in the one-degree polynomial above. Geometrically, the graph of the function must pass through the origin.
