Linearity is the property of a mathematical relationship or function which means that it can be graphically represented as a straight line. Examples are the relationship of voltage and current across a resistor (Ohm's law), or the mass and weight of an object. Proportionality implies linearity, but linearity does not imply proportionality.
1 In mathematics
1.1 Linear polynomials 1.2 Boolean functions
2 Physics 3 Electronics
3.1 Integral linearity
4 Military tactical formations 5 Art 6 Music 7 Measurement 8 See also 9 References 10 External links
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.
f ( x ) = 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 b = 0. Hence, if b ≠ 0, 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
for which there exist
, … ,
0 , 1
displaystyle a_ 0 ,a_ 1 ,ldots ,a_ n in 0,1
, … ,
) ⊕ ⋯ ⊕ (
displaystyle f(b_ 1 ,ldots ,b_ n )=a_ 0 oplus (a_ 1 land b_ 1 )oplus cdots oplus (a_ n land b_ n )
, … ,
0 , 1
displaystyle b_ 1 ,ldots ,b_ n in 0,1 .
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 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, 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 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, 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
Negation, Logical biconditional, exclusive or, tautology, and
contradiction are linear functions.
In physics, linearity is a property of the differential equations
governing many systems; for instance, the
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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, 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.
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
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
Linear is one of the five categories proposed by Swiss art historian
This section needs expansion. You can help by adding to it. (March 2013)
Linear differential equation
^ Edwards, Harold M. (1995). Linear Algebra. Springer. p. 78. ISBN 9780817637316. ^ Stewart, James (2008). Calculus: Early Transcendentals, 6th ed., Brooks Cole Cengage Learning. ISBN 978-0-495-01166-8, Section 1.2 ^ Evans, Lawrence C. (2010) , Partial differential equations (PDF), Graduate Studies in Mathematics, 19 (2nd ed.), Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-4974-3, MR 2597943 ^ Whitaker, Jerry C. (2002). The RF transmission systems handbook. CRC Press. ISBN 978-0-8493-0973-1. ^ Kolts, Bertram S. (2005). "Understanding Linearity and Monotonicity" (PDF). analogZONE. Archived from the original (PDF) on February 4, 2012. Retrieved September 24, 2014. ^ Kolts, Bertram S. (2005). "Understanding Linearity and Monotonicity". Foreign Electronic Measurement Technology. 24 (5): 30–31. Retrieved September 25, 2014. ^ Wölfflin, Heinrich (1950). Hottinger, M.D., ed. Principles of Art History: The Problem of the Development of Style in Later Art. New York: Dover. pp. 18–72.
The dictionary definition of linea