The word linear comes from the Latin word linearis, which means created by lines. Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome. In mathematics, a linear function f(x) is one which satisfies the following two properties:

• Additivity property (also called the superposition property): f(x + y) = f(x) + f(y). Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Physics and Systems theory, the superposition principle, also known as superposition property, states that for all Linear systems This says that f is a group homomorphism with respect to addition. In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function
• Homogeneity property: f(αx) = αf(x) for all α. It turns out that homogeneity follows from the additivity property in all cases where α is rational. In that case if the linear function is continuous, homogeneity is not an additional axiom to establish if the additivity property is established.

In this definition, x is not necessarily a real number, but can in general be a member of any vector space. In Mathematics, the real numbers may be described informally in several different ways In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added

The concept of linearity can be extended to linear operators. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that Important examples of linear operators include the derivative considered as a differential operator, and many constructed from it, such as del and the Laplacian. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Mathematics, a differential operator is an Operator defined as a function of the differentiation operator &nablaDel In Mathematics and Physics, the Laplace operator or Laplacian, denoted by \Delta\  or \nabla^2  and named after When a differential equation can be expressed in linear form, it is particularly easy to solve by breaking the equation up into smaller pieces, solving each of those pieces, and adding the solutions up. A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the

Nonlinear equations and functions are of interest to physicists and mathematicians because they are hard to solve and give rise to interesting phenomena such as chaos. This article describes the use of the term nonlinearity in mathematics A physicist is a Scientist who studies or practices Physics. Physicists study a wide range of physical phenomena in many branches of physics spanning A mathematician is a person whose primary area of study and research is the field of Mathematics. In Mathematics, chaos theory describes the behavior of certain dynamical systems – that is systems whose state evolves with time – that may exhibit dynamics that

Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. Linear algebra is the branch of Mathematics concerned with In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that

See also: linear element, linear system, nonlinearity. In an electric circuit, a linear element is an Electrical element with a Linear relationship between current and Voltage. A linear system is a mathematical model of a System based on the use of a Linear operator. This article describes the use of the term nonlinearity in mathematics

Contents 1 Integral linearity 2 Linear polynomials 3 Boolean functions 4 Physics 5 Electronics 6 Military tactical formations 7 Art 8 Music 9 Measurement 10 References 11 See also

Integral linearity

For a device that converts a quantity to another quantity 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. A multimeter or a multitester, also known as a volt/ohm meter or VOM, is an electronic Measuring instrument that combines several An analog-to-digital converter (abbreviated ADC, A/D or A to D) is an electronic integrated circuit which converts continuous signals to A potentiometer is a three-terminal Resistor with a sliding contact that forms an adjustable Voltage divider. 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. In Electronics, a digital-to-analog converter ( DAC or D-to-A) is a device for converting a digital (usually binary code to an Analog signal

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.

Linear polynomials

In a slightly different usage to the above, a polynomial of degree 1 is said to be linear, because the graph of a function of that form is a line. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations In Mathematics, there are several meanings of degree depending on the subject In mathematics the graph of a function f is the collection of all Ordered pairs ( x, f ( x)

Over the reals, a linear function is one of the form:

f(x) = m x + b

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. In Mathematics, the term linear function can refer to either of two different but related concepts Slope is used to describe the steepness incline gradient or grade of a straight line. In Vector calculus, the gradient of a Scalar field is a Vector field which points in the direction of the greatest rate of increase of the scalar In Coordinate geometry, the y -intercept is the y-value of the point where the Graph of a function or relation intercepts 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 (see in greater generality affine transformation). In Geometry, an affine transformation or affine map or an affinity (from the Latin affinis, "connected with" between two Vector

Boolean functions

In Boolean algebra, a linear function is one such that:

If there exists such that

A Boolean function is linear if A) In every row of the truth table in which the value of the function is 'T', there are 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; or B) 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. Boolean algebra (or Boolean logic) is a logical calculus of truth values, developed by George Boole in the late 1830s A truth table is a Mathematical table used in Logic — specifically in connection with Boolean algebra, Boolean functions and Propositional

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 binary functions. In Logic and Mathematics, negation or not is an operation on Logical values for example the logical value of a Proposition In Logic and Mathematics, logical biconditional (sometimes also known as the material biconditional) is a Logical operator connecting two statements In Classical logic, a contradiction consists of a logical incompatibility between two or more Propositions It occurs when the propositions taken together yield

Physics

In physics, linearity is a property of the differential equations governing a lot of systems (like, for instance Maxwell equations or the diffusion equation). Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric The diffusion equation is a Partial differential equation which describes density fluctuations in a material undergoing Diffusion.

Namely, linearity of a differential equation means that if two functions f and g are solution of the equation, then their sum f+g is also a solution of the equation. A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the

Electronics

In electronics, the linear operating region of a transistor is where the collector-emitter current is related to the base current by a simple scale factor, enabling the transistor to be used as an amplifier that preserves the fidelity of analog signals. Electronics refers to the flow of charge (moving Electrons through Nonmetal conductors (mainly Semiconductors, whereas electrical In Electronics, a transistor is a Semiconductor device commonly used to amplify or switch electronic signals Generally an amplifier or simply amp, is any device that changes usually increases the amplitude of a signal. High fidelity or hi-fi reproduction is a term used by home stereo listeners and home audio enthusiasts ( Audiophiles to refer to high-quality reproduction Linear is similarly used to describe regions of any function, mathematical or physical, that follow a straight line with arbitrary slope.

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. A military organization is a way of structuring the armed forces of a State as a need to offer Military capability required by the National defence policy A pike is a Pole weapon, a very long thrusting Spear used two-handed and used extensively by Infantry both for attacks on enemy foot soldiers and as This kind of formation would get thinner until its extreme in the age of Wellington with the 'Thin Red Line'. The Thin Red Line was a famous military action by the British Army 's 93rd (Highland Regiment at the Battle of Balaclava on October 25, 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. Art history is the Academic study of objects of Art in their Historical development and stylistic contexts i Heinrich Wölfflin ( June 21 1864 &ndash July 19 1945) was a famous Swiss Art critic, whose objective classifying principles The Renaissance (from French Renaissance, meaning "rebirth" Italian: Rinascimento, from re- "again" and nascere Baroque art redirects here Please disambiguate such links to Baroque painting, Baroque sculpture, etc According to Wölfflin, painters of the fifteenth and early sixteenth centuries (Leonardo da Vinci, Raphael or Albrecht Dürer) are more linear than "painterly" Baroque painters of the seventeenth (Peter Paul Rubens, Rembrandt, and Velasquez) because they primarily use outline to create form. Leonardo di ser Piero da Vinci ( April 15 1452 – May 2 1519 was an Italian Polymath, having been a scientist Mathematician, Engineer Raphael Sanzio, usually known by his first name alone (in Italian Raffaello) (April 6 or March 28 1483 – April 6 1520 was an Italian painter and Albrecht Dürer (ˈalbʀɛçt ˈdyʀɐ ( May 21, 1471 &ndash April 6, 1528) was a German painter, Printmaker Painterly is a translation of the German term malerisch, one of the opposed categories popularized by Swiss Art historian Heinrich Wölfflin Rembrandt Harmenszoon van Rijn (July 15 1606 &ndash October 4 1669 was a Dutch painter and etcher. Diego Rodríguez de Silva y Velázquez ( June 6, 1599 &ndash August 6, 1660) was a Spanish painter who was the leading ^[1]

Music

In music the linear aspect is succession, either intervals or melody, as opposed to simultaneity or the vertical aspect. Music is an Art form in which the medium is Sound organized in Time. In Music theory, the term interval describes the relationship between the pitches of two Notes Intervals may be described as vertical In Music, a melody (from Greek μελῳδία - melōidía, "singing chanting" also tune, voice, or In Music theory, the term interval describes the relationship between the pitches of two Notes Intervals may be described as vertical

Measurement

In measurement, the term "linear foot" refers to the number of feet in a straight line of material (such as lumber or fabric) 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". Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome.

References

1. ^ Heinrich Wölfflin, Principles of Art History: the Problem of the Development of Style in Later Art, M. D. Hottinger (trans. ), Mineola, N. Y. : Dover (1950): pp. 18-72.

See also

• Nonlinear
• Linear equation
• Linear medium
• Linear programming
• Bilinear
• Multilinear
• Linear motor
• Linear A and Linear B scripts. This article describes the use of the term nonlinearity in mathematics A linear equation is an Algebraic equation in which each term is either a Constant or the product of a constant and (the first power of a single Variable A linear medium is any medium which is intended to be written to or accessed in a Linear fashion literally meaning in a line. In Mathematics, linear programming (LP is a technique for optimization of a Linear Objective function, subject to Linear equality In Linear algebra, a multilinear map is a Mathematical function of several vector variables that is linear in each variable A linear motor or linear induction motor is essentially a multi-phase Alternating current (AC Electric motor that has had its Stator "unrolled" Linear A is one of two linear scripts used in ancient Crete before Greek Mycenaean Linear B. Linear B is a script that was used for writing Mycenaean, an early form of Greek.
• Linear interpolation

Linear interpolation is a method of Curve fitting using linear polynomials

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
Dapyx Software network: MP3 Explorer | Ebook Manager | Zenithic