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Linear map or polynomial function of degree one
In mathematics, the term linear function refers to two distinct but related notions:
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.
As a linear map
The integral of a function is a linear map from the vector space of integrable functions to the real numbers.
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 above constant b equals zero. Geometrically, the graph of the function must pass through the origin.