Editing Linear function (section)

== As a linear map ==
{{main article|Linear map}}
[[File:Integral as region under curve.svg|thumb|The [[integral]] of a function is a linear map from the vector space of integrable functions to the real numbers.]]

In linear algebra, a linear function is a map ''f'' between two [[vector space]]s such that
:<math>f(\mathbf{x} + \mathbf{y}) = f(\mathbf{x}) + f(\mathbf{y}) </math>
:<math>f(a\mathbf{x}) = af(\mathbf{x}). </math>
Here {{math|''a''}} denotes a constant belonging to some [[field (mathematics)|field]] {{math|''K''}} of [[Scalar (mathematics)|scalar]]s (for example, the [[real number]]s) and {{math|'''x'''}} and {{math|'''y'''}} are elements of a [[vector space]], which might be {{math|''K''}} itself.

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;<ref>Gelfand 1961</ref> these are more commonly called [[linear form]]s.

The "linear functions" of calculus qualify as "linear maps" when (and only when) {{math|1=''f''(0, ..., 0) = 0}}, or, equivalently, when the constant {{mvar|b}} equals zero in the one-degree polynomial above. Geometrically, the graph of the function must pass through the origin.