Linear function

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

$f(x+y)=f(x)+f(y)$

$f(a*x)=a*f(x)$

The linear functions are often confused with affine function. 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

In advanced 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

• Nonlinear system
• Piecewise linear function
• Linear interpolation
• Discontinuous linear map

External links

• Linear Functions on Id Mind
• Interactive tool to explore linear functions