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In mathematics, the term linear function refers to a function that satisfies the following two properties:
Linear functions may be confused with affine functions. One variable affine functions can be written as . Although affine functions make lines when graphed, they do not satisfy the properties of linearity.
Vector spaces 
In mathematics, a linear function means a function that is a linear map, that is, a map between two vector spaces that preserves vector addition and scalar multiplication. For example, if and are represented as coordinate vectors, then the linear functions are those functions that can be expressed as
where M is a matrix. A function
is a linear map if and only if = 0. For other values of this falls in the more general class of affine maps.
Linear functions form the basis of linear algebra.
See also 
- Homogenous function
- Nonlinear system
- Piecewise linear function
- Linear interpolation
- Discontinuous linear map