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Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight
line Line most often refers to: * Line (geometry) In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are One-dimensional space, one-dimensional objects, though they may exist in Two-dimensional Euclide ...
. Linearity is closely related to '' proportionality''. Examples in
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science is that depar ...
include rectilinear motion, the linear relationship of
voltage Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a Electrostatics, static electric field, it corresponds to the Work (electrical), w ...
and current in an electrical conductor (
Ohm's law Ohm's law states that the electric current, current through a Electrical conductor, conductor between two points is directly Proportionality (mathematics), proportional to the voltage across the two points. Introducing the constant of proporti ...
), and the relationship of
mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a body. It was traditionally believed to be related to the physical quantity, quantity of matter in a Physical object, physical body, until the discovery of the atom and par ...
and weight. By contrast, more complicated relationships are ''
nonlinear In mathematics and science, a nonlinear system is a system in which the change of the output is not proportionality (mathematics), proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, m ...
''. Generalized for functions in more than one dimension, linearity means the property of a function of being compatible with addition and scaling, also known as the
superposition principle The superposition principle, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So tha ...
. The word linear comes from
Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through ...
''linearis'', "pertaining to or resembling a line".

# In mathematics

In mathematics, a linear map or
linear function In mathematics, the term linear function refers to two distinct but related notions: * In calculus and related areas, a linear function is a function (mathematics), function whose graph of a function, graph is a straight line, that is, a polynomia ...
''f''(''x'') is a function that satisfies the two properties: * Additivity: . * Homogeneity of degree 1: for all α. These properties are known as the superposition principle. In this definition, ''x'' is not necessarily a
real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
, but can in general be an element of any
vector space In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
. A more special definition of
linear function In mathematics, the term linear function refers to two distinct but related notions: * In calculus and related areas, a linear function is a function (mathematics), function whose graph of a function, graph is a straight line, that is, a polynomia ...
, not coinciding with the definition of linear map, is used in elementary mathematics (see below). Additivity alone implies homogeneity for rational α, since $f\left(x+x\right)=f\left(x\right)+f\left(x\right)$ implies $f\left(nx\right)=n f\left(x\right)$ for any
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
''n'' by mathematical induction, and then $n f\left(x\right) = f\left(nx\right)=f\left(m\tfracx\right)= m f\left(\tfracx\right)$ implies $f\left(\tfracx\right) = \tfrac f\left(x\right)$. The
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek language, Greek letter Rho (letter), rho), although the Latin letter ''D'' ca ...
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 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 In mathematics, a differential equation is an functional equation, equation that relates one or more unknown function (mathematics), functions and their derivatives. In applications, the functions generally represent physical quantities, the der ...
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 algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mat ...
is the branch of mathematics concerned with the study of vectors,
vector space In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
s (also called 'linear spaces'), linear transformations (also called 'linear maps'), and systems of linear equations. For a description of linear and nonlinear equations, see '' linear equation''.

## Linear polynomials

In a different usage to the above definition, a
polynomial In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtrac ...
of degree 1 is said to be linear, because the graph of a function of that form is a straight line. Over the reals, a linear equation is one of the forms: :$f\left(x\right) = m x + b\$ where ''m'' is often called the slope or gradient; ''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 . Hence, if , the function is often called an affine function (see in greater generality affine transformation).

## Boolean functions

In Boolean algebra, a linear function is a function $f$ for which there exist $a_0, a_1, \ldots, a_n \in \$ such that :$f\left(b_1, \ldots, b_n\right) = a_0 \oplus \left(a_1 \land b_1\right) \oplus \cdots \oplus \left(a_n \land b_n\right)$, where $b_1, \ldots, b_n \in \.$ Note that if $a_0 = 1$, 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 A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra (logic), Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expression (mathematics) ...
: # In every row in which the truth value of the function is T, there are an odd number of Ts assigned to the arguments, and in every row in which the function is F there is an even number of Ts assigned to arguments. Specifically, , and these functions correspond to linear maps 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, . 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, and contradiction are linear functions.

# Physics

In
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science is that depar ...
, ''linearity'' is a property of the differential equations governing many systems; for instance, the Maxwell equations or the
diffusion equation The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's laws ...
. Linearity of a homogenous
differential equation In mathematics, a differential equation is an functional equation, equation that relates one or more unknown function (mathematics), functions and their derivatives. In applications, the functions generally represent physical quantities, the der ...
means that if two functions ''f'' and ''g'' are solutions of the equation, then any linear combination 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.

# Electronics

In
electronics The field of electronics is a branch of physics and electrical engineering that deals with the emission, behaviour and effects of electrons using Electronic component, electronic devices. Electronics uses Passivity (engineering), active devices ...
, the linear operating region of a device, for example a
transistor file:MOSFET Structure.png, upright=1.4, Metal-oxide-semiconductor field-effect transistor (MOSFET), showing Metal gate, gate (G), body (B), source (S) and drain (D) terminals. The gate is separated from the body by an insulating layer (pink). A ...
, is where an output dependent variable (such as the transistor collector current) is directly 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 An audio power amplifier (or power amp) is an electronic amplifier that amplifies low-power electronic audio signals, such as the signal from a radio receiver or an electric guitar pickup (music technology), pickup, to a level that is high e ...
, which must amplify a signal without changing its waveform. Others are linear filters, and linear amplifiers in general. In most scientific and 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.

## Integral linearity

For an electronic device (or other physical device) that converts a quantity to another quantity, Bertram S. Kolts writes:
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 In electronics and signal processing, full scale represents the maximum amplitude a system can represent. In digital systems, a signal is said to be at digital full scale when its magnitude has reached the maximum representable value. Once a sig ...
, 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.

# Military tactical formations

In military tactical formations, "linear formations" were adapted starting from phalanx-like formations of pike protected by handgunners, towards shallow formations of handgunners protected by progressively fewer pikes. This kind of formation got progressively thinner until its extreme in the age of Wellington's ' Thin Red Line'. It was eventually replaced by skirmish order when the invention of the breech-loading rifle allowed soldiers to move and fire in small, mobile units, unsupported by large-scale formations of any shape.

# Art

Linear is one of the five categories proposed by Swiss art historian
Heinrich Wölfflin Heinrich Wölfflin (; 21 June 1864 – 19 July 1945) was a Swiss art historian, esthetician and educator, whose objective classifying principles ("painterly" vs. "linear" and the like) were influential in the development of Formalism (art), form ...
to distinguish "Classic", or Renaissance art, from the
Baroque The Baroque (, ; ) is a Style (visual arts), style of Baroque architecture, architecture, Baroque music, music, Baroque dance, dance, Baroque painting, painting, Baroque sculpture, sculpture, poetry, and other arts that flourished in Europe from ...
. According to Wölfflin, painters of the fifteenth and early sixteenth centuries (
Leonardo da Vinci Leonardo di ser Piero da Vinci (15 April 14522 May 1519) was an Italian polymath of the High Renaissance who was active as a painter, Drawing, draughtsman, engineer, scientist, theorist, sculptor, and architect. While his fame initially res ...
,
Raphael Raffaello Sanzio da Urbino, better known as Raphael (; or ; March 28 or April 6, 1483April 6, 1520), was an Italian painter and architect of the High Renaissance. List of works by Raphael, His work is admired for its clarity of form, ease of ...
or Albrecht Dürer) are more linear than "
painterly Painterliness is a concept based on ''german: malerisch'' ('painterly'), a word popularized by Swiss art historian Heinrich Wölfflin (1864–1945) to help focus, enrich and standardize the terms being used by art historians of his time to cha ...
" Baroque painters of the seventeenth century ( Peter Paul Rubens,
Rembrandt Rembrandt Harmenszoon van Rijn (, ; 15 July 1606 – 4 October 1669), usually simply known as Rembrandt, was a Dutch Golden Age painter, printmaker and Drawing, draughtsman. An innovative and prolific Old Masters, master in three art medi ...
, and Velázquez) because they primarily use outline to create
shape A shape or figure is a graphics, graphical representation of an object or its external boundary, outline, or external Surface (mathematics), surface, as opposed to other properties such as color, Surface texture, texture, or material type. A pl ...
. Linearity in art can also be referenced in digital art. For example, hypertext fiction can be an example of nonlinear narrative, but there are also websites designed to go in a specified, organized manner, following a linear path.

# Music

In music the linear aspect is succession, either intervals or
melody A melody (from Greek language, Greek μελῳδία, ''melōidía'', "singing, chanting"), also tune, voice or line, is a Linearity#Music, linear succession of musical tones that the listener perceives as a single entity. In its most liter ...
, as opposed to simultaneity or the vertical aspect.