Linearity is the property of a mathematical relationship (''
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
'') that can be
graphically represented as a straight
line. Linearity is closely related to ''
proportionality''. Examples in
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
include
rectilinear motion
Linear motion, also called rectilinear motion, is one-dimensional motion along a straight line, and can therefore be described mathematically using only one spatial dimension. The linear motion can be of two types: uniform linear motion, with co ...
, 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 static electric field, it corresponds to the work needed per unit of charge to ...
and
current
Currents, Current or The Current may refer to:
Science and technology
* Current (fluid), the flow of a liquid or a gas
** Air current, a flow of air
** Ocean current, a current in the ocean
*** Rip current, a kind of water current
** Current (stre ...
in an
electrical conductor
In physics and electrical engineering, a conductor is an object or type of material that allows the flow of charge (electric current) in one or more directions. Materials made of metal are common electrical conductors. Electric current is gene ...
(
Ohm's law), and the relationship of
mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
and
weight
In science and engineering, the weight of an object is the force acting on the object due to gravity.
Some standard textbooks define weight as a vector quantity, the gravitational force acting on the object. Others define weight as a scalar qua ...
. 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 proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many othe ...
''.
Generalized for functions in more than one
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
, linearity means the property of a function of being compatible with
addition and
scaling
Scaling may refer to:
Science and technology
Mathematics and physics
* Scaling (geometry), a linear transformation that enlarges or diminishes objects
* Scale invariance, a feature of objects or laws that do not change if scales of length, energ ...
, also known as the
superposition principle.
The word linear comes from
Latin
Latin (, or , ) is a classical language belonging to the 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 the power of the ...
''linearis'', "pertaining to or resembling a line".
In mathematics
In mathematics, a
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
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 whose graph is a straight line, that is, a polynomial function of degree zero or one. For dist ...
''f''(''x'') is a function that satisfies the two properties:
*
Additivity
Additive may refer to:
Mathematics
* Additive function, a function in number theory
* Additive map, a function that preserves the addition operation
* Additive set-functionn see Sigma additivity
* Additive category, a preadditive category with f ...
: .
*
Homogeneity
Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the Uniformity (chemistry), uniformity of a Chemical substance, substance or organism. A material or image that is homogeneous is uniform in compos ...
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 measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
, but can in general be an
element of any
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
. 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 whose graph is a straight line, that is, a polynomial function of degree zero or one. For dist ...
, not coinciding with the definition of linear map, is used in elementary mathematics (see below).
Additivity alone implies homogeneity for
rational
Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...
α, since
implies
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'' by
mathematical induction
Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ... all hold. Informal metaphors help ...
, and then
implies
. 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 letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
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
Operator may refer to:
Mathematics
* A symbol indicating a mathematical operation
* Logical operator or logical connective in mathematical logic
* Operator (mathematics), mapping that acts on elements of a space to produce elements of another sp ...
. Important examples of linear operators include the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
considered as a
differential operator, and other operators constructed from it, such as
del
Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a one-dimensional domain, it denotes ...
and the
Laplacian. When a
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
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 matrices ...
is the branch of mathematics concerned with the study of
vectors,
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s (also called 'linear spaces'),
linear transformation
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
s (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 consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
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:
:
where ''m'' is often called the
slope
In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...
or
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
; ''b'' the
y-intercept
In analytic geometry, using the common convention that the horizontal axis represents a variable ''x'' and the vertical axis represents a variable ''y'', a ''y''-intercept or vertical intercept is a point where the graph of a function or relatio ...
, 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
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is b ...
. Hence, if , the function is often called an affine function (see in greater generality
affine transformation).
Boolean functions
![Hasse-linear](https://upload.wikimedia.org/wikipedia/commons/7/73/Hasse-linear.svg)
In
Boolean algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas i ...
, a linear function is a function
for which there exist
such that
:
, where
Note that if
, 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, boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional argumen ...
:
# 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 map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
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
In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values ('' true'' or '' false'').
Computing
In some pro ...
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
In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values ('' true'' or '' false'').
Computing
In some pro ...
of the operation or it never makes a difference.
Negation,
Logical biconditional
In logic and mathematics, the logical biconditional, sometimes known as the material biconditional, is the logical connective (\leftrightarrow) used to conjoin two statements and to form the statement " if and only if ", where is known as th ...
,
exclusive or
Exclusive or or exclusive disjunction is a logical operation that is true if and only if its arguments differ (one is true, the other is false).
It is symbolized by the prefix operator J and by the infix operators XOR ( or ), EOR, EXOR, , ...
,
tautology, and
contradiction
In traditional logic, a contradiction occurs when a proposition conflicts either with itself or established fact. It is often used as a tool to detect disingenuous beliefs and bias. Illustrating a general tendency in applied logic, Aristotle's ...
are linear functions.
Physics
In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, ''linearity'' is a property of the
differential equations governing many systems; for instance, the
Maxwell equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
T ...
or the
diffusion equation.
Linearity of a homogenous
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
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
In neuroscience and psychophysics, an absolute threshold was originally defined as the lowest level of a stimulus – light, sound, touch, etc. – that an organism could detect. Under the influence of signal detection theory, absolute thresho ...
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 devices. Electronics uses active devices to control electron flow by amplification ...
, the linear operating region of a device, for example a
transistor
upright=1.4, gate (G), body (B), source (S) and drain (D) terminals. The gate is separated from the body by an insulating layer (pink).
A transistor is a semiconductor device used to Electronic amplifier, amplify or electronic switch, switch ...
, is where an output
dependent variable (such as the transistor collector
current
Currents, Current or The Current may refer to:
Science and technology
* Current (fluid), the flow of a liquid or a gas
** Air current, a flow of air
** Ocean current, a current in the ocean
*** Rip current, a kind of water current
** Current (stre ...
) 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, which must amplify a signal without changing its waveform. Others are
linear filter
Linear filters process time-varying input signals to produce output signals, subject to the constraint of linearity. In most cases these linear filters are also time invariant (or shift invariant) in which case they can be analyzed exactly using ...
s, and
linear amplifier
A linear amplifier is an electronic circuit whose output is proportional to its input, but capable of delivering more power into a load. The term usually refers to a type of radio-frequency (RF) power amplifier, some of which have output power m ...
s in general.
In most
scientific
Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe.
Science may be as old as the human species, and some of the earliest archeological evidence for ...
and
technological
Technology is the application of knowledge to reach practical goals in a specifiable and reproducible way. The word ''technology'' may also mean the product of such an endeavor. The use of technology is widely prevalent in medicine, science, ...
, 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 si ...
, 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
Pike, Pikes or The Pike may refer to:
Fish
* Blue pike or blue walleye, an extinct color morph of the yellow walleye ''Sander vitreus''
* Ctenoluciidae, the "pike characins", some species of which are commonly known as pikes
* ''Esox'', genus of ...
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 formal analysis in ar ...
to distinguish "Classic", or
Renaissance art
Renaissance art (1350 – 1620 AD) is the painting, sculpture, and decorative arts of the period of European history known as the Renaissance, which emerged as a distinct style in Italy in about AD 1400, in parallel with developments which occ ...
, from the
Baroque. 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. His work is admired for its clarity of form, ease of composition, and visual a ...
or
Albrecht Dürer) are more linear than "
painterly" Baroque painters of the seventeenth century (
Peter Paul Rubens
Sir Peter Paul Rubens (; ; 28 June 1577 – 30 May 1640) was a Flemish artist and diplomat from the Duchy of Brabant in the Southern Netherlands (modern-day Belgium). He is considered the most influential artist of the Flemish Baroque tradi ...
,
Rembrandt, and
Velázquez) because they primarily use outline to create
shape
A shape or figure is a graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type.
A plane shape or plane figure is constrained to lie ...
.
Linearity in art can also be referenced in
digital art
Digital art refers to any artistic work or practice that uses digital technology as part of the creative or presentation process, or more specifically computational art that uses and engages with digital media.
Since the 1960s, various name ...
. For example,
hypertext fiction
Hypertext fiction is a genre of electronic literature, characterized by the use of hypertext links that provide a new context for non-linearity in literature and reader interaction. The reader typically chooses links to move from one node of text ...
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, as opposed to
simultaneity
Simultaneity may refer to:
* Relativity of simultaneity, a concept in special relativity.
* Simultaneity (music), more than one complete musical texture occurring at the same time, rather than in succession
* Simultaneity, a concept in Endogeneit ...
or the
vertical
Vertical is a geometric term of location which may refer to:
* Vertical direction, the direction aligned with the direction of the force of gravity, up or down
* Vertical (angles), a pair of angles opposite each other, formed by two intersecting s ...
aspect.
In statistics
See also
*
Linear actuator
A linear actuator is an actuator that creates motion in a straight line, in contrast to the circular motion of a conventional electric motor. Linear actuators are used in machine tools and industrial machinery, in computer peripherals such as ...
*
Linear element
Electrical elements are conceptual abstractions representing idealized electrical components, such as resistors, capacitors, and inductors, used in the analysis of electrical networks. All electrical networks can be analyzed as multiple electrica ...
*
Linear foot
*
Linear system
In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator.
Linear systems typically exhibit features and properties that are much simpler than the nonlinear case.
As a mathematical abstractio ...
*
Linear programming
*
Linear differential equation
In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form
:a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b ...
*
Bilinear
*
Multilinear
*
Linear motor
*
Linear A
Linear A is a writing system that was used by the Minoans of Crete from 1800 to 1450 BC to write the hypothesized Minoan language or languages. Linear A was the primary script used in palace and religious writings of the Minoan civi ...
and
Linear B scripts.
*
Linear interpolation
In mathematics, linear interpolation is a method of curve fitting using linear polynomials to construct new data points within the range of a discrete set of known data points.
Linear interpolation between two known points
If the two known poi ...
References
External links
*{{wiktionary-inline
Elementary algebra
Physical phenomena
Broad-concept articles