Linear 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 :f(x)=ax+b, 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 f(x_1, \ldots, x_k) of any finite number of variables, the general formula is :f(x_1, \ldots, x_k) = b + a_1 x_1 + \cdots + a_k x_k , 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 (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. == As a linear map ==

of an integrable function is a linear map from a vector space of integrable functions to real numbers (that is also a vector space). In linear algebra, a linear function is a map f from a vector space \mathbf{V} to a vector space \mathbf{W} (Both spaces are not necessarily different.) over a same field such that :f(\mathbf{x} + \mathbf{y}) = f(\mathbf{x}) + f(\mathbf{y}) :f(a\mathbf{x}) = af(\mathbf{x}). Here denotes a constant belonging to the field of scalars (for example, the real numbers), and and are elements of \mathbf{V}, which might be itself. Even if the same symbol + is used, the operation of addition between and (belonging to \mathbf{V}) is not necessarily same to the operation of addition between f\left( \mathbf{x} \right) and f\left( \mathbf{y} \right) (belonging to \mathbf{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; 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. == See also ==