Linear function

For the use of the term in calculus, see Linear function (calculus).

In mathematics, the term linear function refers to two distinct but related notions:

• In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. For distinguishing such a linear function from the other concept, the term affine function is often used.
• In linear algebra, mathematical analysis, and functional analysis, a linear function is a linear map.

As a polynomial function

Main article: Linear function (calculus)

Graphs of two linear functions.

In calculus, analytic geometry and related areas, a linear function is a polynomial of degree one or less,

• the zero polynomial (the latter not being considered to have degree zero).

When the function is of only one variable, it is of the form

$f(x)=ax+b,$

where a and b are constants, often real numbers. The graph of such a function of one variable is a nonvertical line. a is frequently referred to as the slope of the line, and b as the intercept.

For a function $f(x_{1},\ldots ,x_{k})$ 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 k.

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) 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

Main article: Linear map

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 spaces s.t.

$f(\mathbf {x} +\mathbf {y} )=f(\mathbf {x} )+f(\mathbf {y} )$

$f(a\mathbf {x} )=af(\mathbf {x} ).$

Here a denotes a constant belonging to some field K of scalars (for example, the real numbers) and x and y are elements of a vector space, which might be 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; these are more commonly called linear forms.

The "linear functions" of calculus qualify as "linear maps" when (and only when) f(0, ..., 0) = 0, or, equivalently, when the above constant b equals zero. Geometrically, the graph of the function must pass through the origin.

See also

• Homogeneous function
• Nonlinear system
• Piecewise linear function
• Linear approximation
• Linear interpolation
• Discontinuous linear map
• Linear least squares

References

• Izrail Moiseevich Gelfand (1961), Lectures on Linear Algebra, Interscience Publishers, Inc., New York. Reprinted by Dover, 1989. ISBN 0-486-66082-6
• Thomas S. Shores (2007), Applied Linear Algebra and Matrix Analysis, Undergraduate Texts in Mathematics, Springer. ISBN 0-387-33195-6
• James Stewart (2012), Calculus: Early Transcendentals, edition 7E, Brooks/Cole. ISBN 978-0-538-49790-9
• Leonid N. Vaserstein (2006), "Linear Programming", in Leslie Hogben, ed., Handbook of Linear Algebra, Discrete Mathematics and Its Applications, Chapman and Hall/CRC, chap. 50. ISBN 1-584-88510-6