Linear function

Inmathematics,the termlinear functionrefers to two distinct but related notions:^[1]

Incalculusand related areas, a linear function is afunctionwhosegraphis astraight line,that is, apolynomial functionofdegreezero or one.^[2]For distinguishing such a linear function from the other concept, the termaffine functionis often used.^[3]
Inlinear algebra,mathematical analysis,^[4]andfunctional analysis,a linear function is alinear map.^[5]

As a polynomial function[edit]

In calculus,analytic geometryand related areas, a linear function is a polynomial of degree one or less, including thezero polynomial(the latter not being considered to have degree zero).

When the function is of only onevariable,it is of the form

f(x)=ax+b,

where $a$ and $b$ areconstants,oftenreal numbers.Thegraphof 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.

Ifa > 0then thegradientis positive and the graph slopes upwards.

Ifa < 0then thegradientis negative and the graph slopes downwards.

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 ahyperplaneof dimensionk.

Aconstant functionis 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 ahomogeneouslinear function or alinear form.In the context of linear algebra, the polynomial functions of degree 0 or 1 are the scalar-valuedaffine maps.

As a linear map[edit]

In linear algebra, a linear function is a mapfbetween twovector spacessuch that

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 somefield $K$ ofscalars(for example, thereal numbers) and $x$ and $y$ are elements of avector space,which might be $K$ itself.

In other terms the linear function preservesvector additionandscalar multiplication.

Some authors use "linear function" only for linear maps that take values in the scalar field;^[6]these are more commonly calledlinear forms.

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

Notes[edit]

^"The termlinear functionmeans a linear form in some textbooks and an affine function in others. "Vaserstein 2006, p. 50-1
^Stewart 2012, p. 23
^A. Kurosh (1975).Higher Algebra.Mir Publishers. p. 214.
^T. M. Apostol (1981).Mathematical Analysis.Addison-Wesley. p. 345.
^Shores 2007, p. 71
^Gelfand 1961

References[edit]

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", inLeslie Hogben,ed.,Handbook of Linear Algebra,Discrete Mathematics and Its Applications, Chapman and Hall/CRC, chap. 50.ISBN 1-584-88510-6

[1] "The termlinear functionmeans a linear form in some textbooks and an affine function in others. "Vaserstein 2006, p. 50-1

[2] Stewart 2012, p. 23

[3] A. Kurosh (1975).Higher Algebra.Mir Publishers. p. 214.

[4] T. M. Apostol (1981).Mathematical Analysis.Addison-Wesley. p. 345.

[5] Shores 2007, p. 71

[6] Gelfand 1961

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As a polynomial function[edit]

As a linear map[edit]

See also[edit]

Notes[edit]

References[edit]