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{{Short description|Properties of mathematical relationships}}
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{{Distinguish|Lineage (disambiguation){{!}}Lineage}}
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In mathematics, the term '''''linear''''' is used in two distinct senses for two different properties:
* linearity of a ''[[function (mathematics)|function]]'' (or ''[[mapping  (mathematics)|mapping]]'');
* linearity of a ''[[polynomial]]''.
An example of a linear function is the function defined by <math>f(x)=(ax,bx)</math> that maps the real line to a line in the [[Euclidean plane]] '''R'''<sup>2</sup> that passes through the origin. An example of a linear polynomial in the variables <math>X,</math> <math>Y</math> and <math>Z</math> is <math>aX+bY+cZ+d.</math> 

Linearity of a mapping is closely related to ''[[Proportionality (mathematics)|proportionality]]''. Examples in [[physics]] include the linear relationship of [[voltage]] and [[Electric current|current]] in an [[electrical conductor]] ([[Ohm's law]]), and the relationship of [[mass]] and [[weight]]. By contrast, more complicated relationships, such as between [[velocity]] and [[kinetic energy]], are ''[[Nonlinear system|nonlinear]]''.

Generalized for functions in more than one [[dimension (mathematics)|dimension]], linearity means the property of a function of being compatible with [[addition]] and [[scale analysis (mathematics)|scaling]], also known as the [[superposition principle]].

Linearity of a polynomial means that its [[degree of a polynomial|degree]] is less than two. The use of the term for polynomials stems from the fact that the [[graph of a function|graph]] of a polynomial in one variable is a straight [[line (geometry)|line]]. In the term "[[linear equation]]", the word refers to the linearity of the polynomials involved.

Because a function such as <math>f(x)=ax+b</math> is defined by a linear polynomial in its argument, it is sometimes also referred to as being a "linear function", and the relationship between the argument and the function value may be referred to as a "linear relationship". This is potentially confusing, but usually the intended meaning will be clear from the context.

The word '''linear''' comes from [[Latin]] ''linearis'', "pertaining to or resembling a line".

==In mathematics==

===Linear maps===
In mathematics, a [[linear map]] or [[linear function]] ''f''(''x'') is a function that satisfies the two properties:<ref>{{cite book|author=Edwards, Harold M.|title=Linear Algebra|publisher=Springer|year=1995|isbn=9780817637316|page=78|url=https://books.google.com/books?id=ylFR4h5BIDEC&pg=PA78}}</ref>

* [[Additive map|Additivity]]: {{nowrap|1=''f''(''x'' + ''y'') = ''f''(''x'') + ''f''(''y'')}}.
* [[Homogeneous function|Homogeneity]] of degree 1: {{nowrap|1=''f''(α''x'') = α ''f''(''x'')}} for all α.

These properties are known as the [[superposition principle]]. In this definition, ''x'' is not necessarily a [[real number]], but can in general be an [[element (mathematics)|element]] of any [[vector space]]. A more special definition of [[linear function#As a polynomial function|linear function]], not coinciding with the definition of linear map, is used in elementary mathematics (see below).

Additivity alone implies homogeneity for [[Rational number|rational]] α, since <math>f(x+x)=f(x)+f(x)</math> implies <math>f(nx)=n f(x)</math> for any [[natural number]] ''n'' by [[mathematical induction]], and then <math>n f(x) = f(nx)=f(m\tfrac{n}{m}x)= m f(\tfrac{n}{m}x)</math> implies <math>f(\tfrac{n}{m}x) = \tfrac{n}{m} f(x)</math>. The [[Dense set|density]] of the rational numbers in the reals implies that any additive [[continuous function]] is homogeneous for any real number α, and is therefore linear.

The concept of linearity can be extended to linear [[Operator (mathematics)|operators]]. Important examples of linear operators include the [[derivative]] considered as a [[differential operator]], and other operators constructed from it, such as [[del]] and the [[Laplacian]]. When a [[differential equation]] can be expressed in linear form, it can generally be solved by breaking the equation up into smaller pieces, solving each of those pieces, and summing the solutions.

===Linear polynomials===
{{main|Linear equation|Linear algebra}}
In a different usage to the above definition, a [[polynomial]] of degree 1 is said to be linear, because the [[graph of a function]] of that form is a straight line.<ref>[[James Stewart (mathematician)|Stewart, James]] (2008). ''Calculus: Early Transcendentals'', 6th ed., Brooks Cole Cengage Learning. {{isbn|978-0-495-01166-8}}, Section 1.2</ref> 

Over the reals, a simple example of a [[linear equation]] is given by:

:<math>y = m x + b\ </math>

where ''m'' is often called the [[slope]] or [[gradient]], and ''b'' the [[y-intercept]], which gives the point of intersection between the graph of the function and the ''y''-axis.

Note that this usage of the term ''linear'' is not the same as in the section above, because linear polynomials over the real numbers do not in general satisfy either additivity or homogeneity. In fact, they do so [[if and only if]] the [[constant term]] – ''b'' in the example – equals 0. If {{nowrap|''b'' ≠ 0}}, the function is called an '''affine function''' (see in greater generality [[affine transformation]]).

[[Linear algebra]] is the branch of mathematics concerned with systems of linear equations.

===Boolean functions===
{{main article|Parity function}}
[[File:Hasse-linear.svg|thumb|right|Hasse diagram of a linear Boolean function]]
In [[Boolean algebra (logic)|Boolean algebra]], a linear function is a function <math>f</math> for which there exist <math>a_0, a_1, \ldots, a_n \in \{0,1\}</math> such that
:<math>f(b_1, \ldots, b_n) = a_0 \oplus (a_1 \land b_1) \oplus \cdots \oplus (a_n \land b_n)</math>, where <math>b_1, \ldots, b_n \in \{0,1\}.</math>

Note that if <math>a_0 = 1</math>, the above function is considered affine in linear algebra (i.e. not linear).

A Boolean function is linear if one of the following holds for the function's [[truth table]]:
# In every row in which the truth value of the function is [[Truth value#Classical logic|T]], there are an odd number of Ts assigned to the arguments, and in every row in which the function is [[Truth value#Classical logic|F]] there is an even number of Ts assigned to arguments. Specifically, {{nowrap|1=''f''(F, F, ..., F) = F}}, and these functions correspond to [[linear map]]s over the Boolean vector space.
# In every row in which the value of the function is T, there is an even number of Ts assigned to the arguments of the function; and in every row in which the [[truth value]] of the function is F, there are an odd number of Ts assigned to arguments. In this case, {{nowrap|1=''f''(F, F, ..., F) = T}}.

Another way to express this is that each variable always makes a difference in the [[truth value]] of the operation or it never makes a difference.

[[Negation]], [[Logical biconditional]], [[exclusive or]], [[tautology (logic)|tautology]], and [[contradiction]] are linear functions.

==Physics==
{{main |Superposition principle}}
In [[physics]], ''linearity'' is a property of the [[differential equation]]s governing many systems; for instance, the [[Maxwell equations]] or the [[diffusion equation]].<ref>{{Citation | last1=Evans | first1=Lawrence C. | title=Partial differential equations | orig-year=1998 | url=https://www.ams.org/journals/bull/2000-37-03/S0273-0979-00-00868-5/S0273-0979-00-00868-5.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://www.ams.org/journals/bull/2000-37-03/S0273-0979-00-00868-5/S0273-0979-00-00868-5.pdf |archive-date=2022-10-09 |url-status=live | publisher=[[American Mathematical Society]] | location=Providence, R.I. | edition=2nd | series=[[Graduate Studies in Mathematics]] | isbn=978-0-8218-4974-3 |mr=2597943 | year=2010 | volume=19 | doi=10.1090/gsm/019}}</ref>

Linearity of a homogenous differential equation means that if two functions ''f'' and ''g'' are solutions of the equation, then any [[linear combination]] {{nowrap|''af'' + ''bg''}} is, too.

In instrumentation, linearity means that a given change in an input variable gives the same change in the output of the measurement apparatus: this is highly desirable in scientific work. In general, instruments are close to linear over a certain range, and most useful within that range. In contrast, human senses are highly nonlinear: for instance, the brain completely ignores incoming light unless it exceeds a certain [[absolute threshold]] number of photons.

[[Linear motion]] traces a straight line trajectory.

==Electronics==
In [[electronics]], the linear operating region of a device, for example a [[transistor]], is where an output [[dependent variable]] (such as the transistor collector [[Electric current|current]]) is directly [[Proportionality (mathematics)|proportional]] to an input dependent variable (such as the base current). This ensures that an analog output is an accurate representation of an input, typically with higher amplitude (amplified). A typical example of linear equipment is a [[high fidelity]] [[audio amplifier]], which must amplify a signal without changing its waveform. Others are [[linear filter]]s, and [[linear amplifier]]s in general.

In most [[Science|scientific]] and [[Technology|technological]], as distinct from mathematical, applications, something may be described as linear if the characteristic is approximately but not exactly a straight line; and linearity may be valid only within a certain operating region—for example, a high-fidelity amplifier may distort a small signal, but sufficiently little to be acceptable (acceptable but imperfect linearity); and may distort very badly if the input exceeds a certain value.<ref name=Whitaker>{{cite book|last=Whitaker|first=Jerry C.|title=The RF transmission systems handbook|year=2002|publisher=CRC Press|isbn=978-0-8493-0973-1|url=https://books.google.com/books?id=G5UHVIqEWdQC&pg=SA11-PA1}}</ref>

===Integral linearity===
{{main |Integral linearity}}
For an electronic device (or other physical device) that converts a quantity to another quantity, Bertram S. Kolts writes:<ref>{{cite web |title=Understanding Linearity and Monotonicity |first=Bertram S. |last=Kolts |publisher=analogZONE |date=2005 |url=http://www.analogzone.com/nett1108.pdf |archive-url=https://web.archive.org/web/20120204065155/http://www.analogzone.com/nett1108.pdf |archive-date=February 4, 2012 |access-date=September 24, 2014}}</ref><ref>{{cite journal |title=Understanding Linearity and Monotonicity |first=Bertram S. |last=Kolts |journal=Foreign Electronic Measurement Technology |year=2005 |volume=24 |issue=5 |pages=30–31 |url=http://caod.oriprobe.com/articles/9294129/Understanding_Linearity_and_Monotonicity.htm |access-date=September 25, 2014}}</ref>

<blockquote>There are three basic definitions for integral linearity in common use: independent linearity, zero-based linearity, and terminal, or end-point, linearity. In each case, linearity defines how well the device's actual performance across a specified operating range approximates a straight line. Linearity is usually measured in terms of a deviation, or non-linearity, from an ideal straight line and it is typically expressed in terms of percent of [[full scale]], or in ppm (parts per million) of full scale. Typically, the straight line is obtained by performing a least-squares fit of the data. The three definitions vary in the manner in which the straight line is positioned relative to the actual device's performance. Also, all three of these definitions ignore any gain, or offset errors that may be present in the actual device's performance characteristics.

<br /></blockquote>

==See also==
*[[Linear actuator]]
*[[Linear element]]
*[[Linear foot]]
*[[Linear system]]
*[[Linear programming]]
*[[Linear differential equation]]
*[[Bilinear form|Bilinear]]
*[[Multilinear form|Multilinear]]
*[[Linear motor]]
*[[Linear interpolation]]

==References==
{{reflist}}

==External links==
*{{wiktionary-inline}}

[[Category:Physical phenomena]]
[[Category:Broad-concept articles]]