Linear function

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In mathematics, the term linear function refers to a function that satisfies the following two properties:

f(x + y) = f(x) + f(y)
f(ax) = af(x).

Linear functions may be confused with affine functions. One variable affine functions can be written as f(x) = mx + b. Although affine functions make lines when graphed, they do not satisfy the properties of linearity.

Vector spaces [edit]

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 x and f(x) are represented as coordinate vectors, then the linear functions are those functions f that can be expressed as

f(x) = \mathrm{M}x,

where M is a matrix. A function

f(x) = mx + b

is a linear map if and only if b = 0. For other values of b this falls in the more general class of affine maps.

Linear functions form the basis of linear algebra.

See also [edit]

External links [edit]