1677:, 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.
1984:
1700:, 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.
50:
1124:
1611:
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,
488:
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
1695:
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.
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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
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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
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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.
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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
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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
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For an electronic device (or other physical device) that converts a quantity to another quantity, Bertram S. Kolts writes:
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human senses are highly nonlinear: for instance, the brain completely ignores incoming light unless it exceeds a certain
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1366:{\displaystyle f(b_{1},\ldots ,b_{n})=a_{0}\oplus (a_{1}\land b_{1})\oplus \cdots \oplus (a_{n}\land b_{n})}
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622:, not coinciding with the definition of linear map, is used in elementary mathematics (see below).
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of the function is F, there are an odd number of Ts assigned to arguments. In this case,
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Another way to express this is that each variable always makes a difference in the
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is less than two. The use of the term for polynomials stems from the fact that the
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that passes through the origin. An example of a linear polynomial in the variables
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A Boolean function is linear if one of the following holds for the function's
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Linearity of a homogenous differential equation means that if two functions
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is the branch of mathematics concerned with systems of linear equations.
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1658:, which must amplify a signal without changing its waveform. Others are
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410:, linearity means the property of a function of being compatible with
858:{\displaystyle nf(x)=f(nx)=f(m{\tfrac {n}{m}}x)=mf({\tfrac {n}{m}}x)}
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there is an even number of Ts assigned to arguments. Specifically,
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1050:
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441:", the word refers to the linearity of the polynomials involved.
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941:
is homogeneous for any real number α, and is therefore linear.
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of the rational numbers in the reals implies that any additive
389:. By contrast, more complicated relationships, such as between
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43:
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is used in two distinct senses for two different properties:
181:
An example of a linear function is the function defined by
1634:, the linear operating region of a device, for example a
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In every row in which the truth value of the function is
926:{\displaystyle f({\tfrac {n}{m}}x)={\tfrac {n}{m}}f(x)}
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948:. Important examples of linear operators include the
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1225:{\displaystyle a_{0},a_{1},\ldots ,a_{n}\in \{0,1\}}
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944:The concept of linearity can be extended to linear
531:) is a function that satisfies the two properties:
74:. Unsourced material may be challenged and removed.
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1528:of the operation or it never makes a difference.
986:In a different usage to the above definition, a
1433:{\displaystyle b_{1},\ldots ,b_{n}\in \{0,1\}.}
990:of degree 1 is said to be linear, because the
433:of a polynomial in one variable is a straight
355:Linearity of a mapping is closely related to
30:"Linear" redirects here. For other uses, see
8:
1424:
1412:
1219:
1207:
406:Generalized for functions in more than one
1950:"Understanding Linearity and Monotonicity"
1917:"Understanding Linearity and Monotonicity"
1576:governing many systems; for instance, the
1127:Hasse diagram of a linear Boolean function
1954:Foreign Electronic Measurement Technology
1809:, 6th ed., Brooks Cole Cengage Learning.
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625:Additivity alone implies homogeneity for
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425:Linearity of a polynomial means that its
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234:that maps the real line to a line in the
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134:Learn how and when to remove this message
1595:are solutions of the equation, then any
503:, "pertaining to or resembling a line".
27:Properties of mathematical relationships
1768:
997:Over the reals, a simple example of a
7:
1891:The RF transmission systems handbook
1502:, and these functions correspond to
72:adding citations to reliable sources
1622:traces a straight line trajectory.
365:include the linear relationship of
1642:(such as the transistor collector
1135:, a linear function is a function
994:of that form is a straight line.
598:These properties are known as the
25:
1072:Note that this usage of the term
1982:
687:{\displaystyle f(x+x)=f(x)+f(x)}
48:
1877:from the original on 2022-10-09
1838:Graduate Studies in Mathematics
1807:Calculus: Early Transcendentals
618:. A more special definition of
59:needs additional citations for
1830:Partial differential equations
1506:over the Boolean vector space.
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1987:The dictionary definition of
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1922:. analogZONE. Archived from
1827:Evans, Lawrence C. (2010) ,
1737:Linear differential equation
1095:, the function is called an
227:{\displaystyle f(x)=(ax,bx)}
1888:Whitaker, Jerry C. (2002).
1776:Edwards, Harold M. (1995).
1099:(see in greater generality
737:{\displaystyle f(nx)=nf(x)}
610:, but can in general be an
444:Because a function such as
381:), and the relationship of
346:{\displaystyle aX+bY+cZ+d.}
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1948:Kolts, Bertram S. (2005).
1915:Kolts, Bertram S. (2005).
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481:{\displaystyle f(x)=ax+b}
146:In mathematics, the term
1782:. Springer. p. 78.
1035:{\displaystyle y=mx+b\ }
489:clear from the context.
37:Not to be confused with
1560:Superposition principle
1469:{\displaystyle a_{0}=1}
600:superposition principle
420:superposition principle
32:Linear (disambiguation)
1703:
1574:differential equations
1550:are linear functions.
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1155:for which there exist
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602:. In this definition,
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954:differential operator
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418:, also known as the
375:electrical conductor
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185:
68:improve this article
18:Linear (mathematics)
1929:on February 4, 2012
1616:number of photons.
992:graph of a function
939:continuous function
2012:Physical phenomena
2007:Elementary algebra
1732:Linear programming
1687:Integral linearity
1681:Integral linearity
1640:dependent variable
1614:absolute threshold
1597:linear combination
1582:diffusion equation
1518:(F, F, ..., F) = T
1500:(F, F, ..., F) = F
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257:{\displaystyle X,}
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1901:978-0-8493-0973-1
1859:978-0-8218-4974-3
1815:978-0-495-01166-8
1664:linear amplifiers
1578:Maxwell equations
1148:{\displaystyle f}
1113:Boolean functions
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296:{\displaystyle Z}
276:{\displaystyle Y}
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1548:contradiction
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575:of degree 1:
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124:December 2007
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85: –
84:
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79:Find sources:
73:
69:
63:
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57:This article
55:
51:
46:
45:
40:
33:
19:
1989:
1962:. Retrieved
1957:
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1931:. Retrieved
1924:the original
1910:
1890:
1883:
1829:
1822:
1806:
1798:
1778:
1771:
1752:Linear motor
1694:
1690:
1668:
1666:in general.
1648:proportional
1629:
1618:
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1600:
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1569:
1563:
1540:exclusive or
1530:
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90:
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66:Please help
61:verification
58:
1747:Multilinear
1722:Linear foot
1632:electronics
1626:Electronics
1526:truth value
1511:truth value
1504:linear maps
1481:truth table
1063:y-intercept
755:, and then
608:real number
573:Homogeneity
512:Linear maps
496:comes from
83:"Linearity"
2001:Categories
1960:(5): 30–31
1763:References
1698:full scale
1671:scientific
1636:transistor
1232:such that
988:polynomial
950:derivative
594:for all α.
536:Additivity
518:linear map
174:polynomial
94:newspapers
1990:linearity
1608:is, too.
1570:linearity
1544:tautology
1410:∈
1394:…
1348:∧
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1329:⋯
1326:⊕
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1189:…
964:. When a
962:Laplacian
946:operators
629:α, since
492:The word
408:dimension
400:nonlinear
379:Ohm's law
1872:archived
1805:(2008).
1742:Bilinear
1706:See also
1669:In most
1532:Negation
1373:, where
1055:gradient
960:and the
865:implies
744:for any
694:implies
627:rational
501:linearis
412:addition
391:velocity
159:function
1868:2597943
1644:current
1580:or the
1566:physics
1554:Physics
1069:-axis.
935:density
614:of any
612:element
416:scaling
371:current
367:voltage
363:physics
165:mapping
108:scholar
39:Lineage
1898:
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1856:
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1786:
1662:, and
1546:, and
1074:linear
1057:, and
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494:linear
427:degree
397:, are
387:weight
373:in an
149:linear
110:
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1927:(PDF)
1920:(PDF)
1875:(PDF)
1834:(PDF)
1051:slope
498:Latin
431:graph
115:JSTOR
101:books
1966:2014
1935:2014
1896:ISBN
1854:ISBN
1811:ISBN
1784:ISBN
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