In common usage, linearity refers to a mathematical relationship or function
that can be graphically represented as a straight line, as in two quantities that
are directly proportional to each other, such as voltage and current in an RLC
circuit, or the mass and weight of an object.
Example A crude but simple example of this
concept can be observed in the volume control of an audio amplifier. While our
ears may perceive a relatively even gradation of volume as the control goes
from 1 to 10, the electrical power consumed in the speaker is rising
geometrically with each numerical increment. The "loudness" is
proportional to the volume number, while the wattage is doubling with every unit
increase. In mathematics
In mathematics, a linear map or linear function f(x) is a function that
satisfies the following two properties: Additivity: f(x + y) = f(x) + f(y).
Homogeneity of degree 1: f(αx) = αf(x) for all α.
The homogeneity and additivity properties together are called the
superposition principle. It can be shown that additivity implies homogeneity in
all cases where α is rational; this is done by proving the case where α is a
natural number by mathematical induction and then extending the result to
arbitrary rational numbers. If f is assumed to be continuous as well, then
this can be extended to show homogeneity for any real number α, using the fact
that rationals form a dense subset of the reals.
In this definition, x is not necessarily a real number, but can in general be a
member of any vector space. A more specific definition of linear function,
not coinciding with the definition of linear map, is used in elementary
mathematics. The concept of linearity can be extended
to linear operators. Important examples of linear operators include the
derivative considered as a differential operator, and many constructed from it,
such as del and the Laplacian. When a differential equation can be expressed
in linear form, it is generally straightforward to solve by breaking the
equation up into smaller pieces, solving each of those pieces, and summing the
solutions. Linear algebra is the branch of
mathematics concerned with the study of vectors, vector spaces, linear
transformations, and systems of linear equations.
The word linear comes from the Latin word linearis, which means pertaining to
or resembling a line. For a description of linear and nonlinear equations, see
linear equation. Nonlinear equations and functions are of interest to physicists
and mathematicians because they can be used to represent many natural
phenomena, including chaos. = Linear polynomials =
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 line.
Over the reals, a linear equation is one of the forms:
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 the 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 b = 0. Hence, if b
≠ 0, the function is often called an affine function.
= Boolean functions = In Boolean algebra, a linear function is
a function for which there exist such that
, where 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 'T', there are an odd
number of 'T's assigned to the arguments and in every row in which the function
is 'F' there is an even number of 'T's assigned to arguments. Specifically,
f('F', 'F', ..., 'F') = 'F', 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 'T's 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 'T's assigned to arguments. In this case, 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, and contradiction are linear functions.
Physics In physics, linearity is a property of
the differential equations governing many systems; for instance, the Maxwell
equations or the diffusion equation. Linearity of a differential equation
means that if two functions f and g are solutions of the equation, then any
linear combination af + bg is, too. Electronics
In electronics, the linear operating region of a device, for example a
transistor, is where a dependent variable is directly proportional to an
independent variable. This ensures that an analog output is an accurate
representation of an input, typically with higher amplitude. A typical example
of linear equipment is a high fidelity audio amplifier, which must amplify a
signal without changing its waveform. Others are linear filters, linear
regulators, 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 even a small signal, but sufficiently little to be acceptable;
and may distort very badly if the input exceeds a certain value, taking it away
from the approximately linear part of the transfer function.
= Integral linearity = For an electronic 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, or in ppm 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. Many times a device's specifications
will simply refer to linearity, with no other explanation as to which type of
linearity is intended. In cases where a specification is expressed simply as
linearity, it is assumed to imply independent linearity.
Independent linearity is probably the most commonly used linearity definition
and is often found in the specifications for DMMs and ADCs, as well as devices
like potentiometers. Independent linearity is defined as the maximum
deviation of actual performance relative to a straight line, located such that it
minimizes the maximum deviation. In that case there are no constraints placed
upon the positioning of the straight line and it may be wherever necessary to
minimize the deviations between it and the device's actual performance
characteristic. Zero-based linearity forces the lower
range value of the straight line to be equal to the actual lower range value of
the device's characteristic, but it does allow the line to be rotated to minimize
the maximum deviation. In this case, since the positioning of the straight
line is constrained by the requirement that the lower range values of the line
and the device's characteristic be coincident, the non-linearity based on
this definition will generally be larger than for independent linearity.
For terminal linearity, there is no flexibility allowed in the placement of
the straight line in order to minimize the deviations. The straight line must
be located such that each of its end-points coincides with the device's
actual upper and lower range values. This means that the non-linearity
measured by this definition will typically be larger than that measured
by the independent, or the zero-based linearity definitions. This definition
of linearity is often associated with ADCs, DACs and various sensors.
A fourth linearity definition, absolute linearity, is sometimes also
encountered. Absolute linearity is a variation of terminal linearity, in that
it allows no flexibility in the placement of the straight line, however
in this case the gain and offset errors of the actual device are included in the
linearity measurement, making this the most difficult measure of a device's
performance. For absolute linearity the end points of the straight line are
defined by the ideal upper and lower range values for the device, rather than
the actual values. The linearity error in this instance is the maximum
deviation of the actual device's performance from ideal.
Military tactical formations In military tactical formations, "linear
formations" were adapted from phalanx-like formations of pike
protected by handgunners towards shallow formations of handgunners protected by
progressively fewer pikes. This kind of formation would get thinner until its
extreme in the age of Wellington with the 'Thin Red Line'. It would eventually
be replaced by skirmish order at the time of the invention of the
breech-loading rifle that allowed soldiers to move and fire independently
of the large-scale formations and fight in small, mobile units.
Art Linear is one of the five categories
proposed by Swiss art historian Heinrich Wölfflin to distinguish "Classic", or
Renaissance art, from the Baroque. According to Wölfflin, painters of the
fifteenth and early sixteenth centuries are more linear than "painterly" Baroque
painters of the seventeenth century because they primarily use outline to
create shape. 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,
as opposed to simultaneity or the vertical aspect.
Measurement In measurement, the term "linear foot"
refers to the number of feet in a straight line of material generally
without regard to the width. It is sometimes incorrectly referred to as
"lineal feet"; however, "lineal" is typically reserved for usage when
referring to ancestry or heredity.[1] The words "linear"[2] & "lineal" [3]
both descend from the same root meaning, the Latin word for line, which is
"linea". See also
Linear actuator Linear element
Linear system Linear medium
Linear programming Linear differential equation
Bilinear Multilinear
Linear motor Linear A and Linear B scripts.
Linear interpolation References
External links The dictionary definition of linearity
at Wiktionary