For other uses, see Linear (disambiguation). The word **linear** comes from the Latin word *linearis*, which means *created by lines*. In mathematics, a **linear** function *f*(*x*) is one which satisfies the following two properties (but see below for a slightly different usage of the term): Look up linear in Wiktionary, the free dictionary. ...
For other uses, see Latin (disambiguation). ...
Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...
Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A...
- Additivity property (also called the superposition property):
*f*(*x* + *y*) = *f*(*x*) + *f*(*y*). This says that *f* is a group homomorphism with respect to addition. - Homogeneity property:
*f*(α*x*) = α*f*(*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 linear algebra, the principle of superposition states that, for a linear system, a linear combination of solutions to the system is also a solution to the same linear system. ...
Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element...
In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...
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. 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 particularly easy to solve by breaking the equation up into smaller pieces, solving each of those pieces, and adding the solutions up. In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...
For a non-technical overview of the subject, see Calculus. ...
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. ...
In vector calculus, del is a vector differential operator represented by the nabla symbol: âˆ‡. Del is a mathematical tool serving primarily as a convention for mathematical notation; it makes many equations easier to comprehend, write, and remember. ...
In vector calculus, the Laplace operator or Laplacian is a differential operator equal to the sum of all the unmixed second partial derivatives of a dependent variable. ...
A simulation of airflow into a duct using the Navier-Stokes equations A differential equation is a mathematical equation for an unknown function of one or several variables which relates the values of the function itself and of its derivatives of various orders. ...
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. To do: 20th century mathematics chaos theory, fractals Lyapunov stability and non-linear control systems non-linear video editing See also: Aleksandr Mikhailovich Lyapunov Dynamical system External links http://www. ...
Not to be confused with physician, a person who practices medicine. ...
Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ...
For other uses, see Chaos Theory (disambiguation). ...
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 the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ...
In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...
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 model of a system based on some kind of linear operator. ...
In mathematics, a nonlinear system is one whose behavior cant be expressed as a sum of the behaviors of its parts (or of their multiples. ...
~~== ~~**==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. 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.~~ A digital multimeter A low cost digital multimeter An analog benchtop multimeter A multimeter or a multitester is an electronic measuring instrument that combines several functions in one unit. ...
4-channel stereo multiplexed analog-to-digital converter WM8775SEDS made by Wolfson Microelectronics placed on X-Fi Fatal1ty Pro sound card An analog-to-digital converter (abbreviated ADC, A/D or A to D) is an electronic integrated circuit (i/c) that converts continuous signals to discrete digital numbers. ...
It has been suggested that Determining emf of primary cells using potentiometer be merged into this article or section. ...
~~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 (current, voltage or electric charge). ...
~~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.~~ ~~== ~~ **Bold text**
==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 that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ...
This article is about the term degree as used in mathematics. ...
relation graph theory In mathematics, the graph of a function f is the collection of all ordered pairs (x,f(x)). In particular, graph means the graphical representation of this collection, in the form of a curve or surface, together with axes, etc. ...
Over the reals, a linear function is one of the form: A linear function is a mathematical function term of the form: f(x) = m x + c where c is a constant. ...
*f*(*x*) = *m* *x* + *b*= Line *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. Look up Slope in Wiktionary, the free dictionary. ...
For other uses, see Gradient (disambiguation). ...
The y-intercept in 2-dimensional space is the point where the graph of a function or relationship intercepts the y-axis of the coordinate system. ...
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). It has been suggested that this article or section be merged into Logical biconditional. ...
In geometry, an affine transformation or affine map (from the Latin, affinis, connected with) between two vector spaces (strictly speaking, two affine spaces) consists of a linear transformation followed by a translation: In the finite-dimensional case each affine transformation is given by a matrix A and a vector b...
## Boolean functions
In Boolean algebra, a linear function is one such that: Boolean algebra is the finitary algebra of two values. ...
If there exists a_{0}, a_{1}, ... , a_{n} {0,1} such that f(b_{1}, ... , b_{n}) = a_{0} ⊕ (a_{1} b_{1}) ⊕ ... ⊕ (a_{n} b_{n}), for all b_{1}, ... , b_{n} {0,1}. 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. Truth tables are a type of mathematical table used in logic to determine whether an expression is true or whether an argument is valid. ...
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. Negation (i. ...
In logical calculus of mathematics, logical biconditional is a logical operator connecting two statements to assert, p if and only if q where p is a hypothesis (or antecedent) and q is a conclusion (or consequent). ...
Exclusive disjunction (usual symbol xor) is a logical operator that results in true if one of the operands (not both) is true. ...
Look up tautology in Wiktionary, the free dictionary. ...
Broadly speaking, a contradiction is an incompatibility between two or more statements, ideas, or actions. ...
## 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). A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ...
In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ...
Maxwells equations are the set of four equations, attributed to James Clerk Maxwell, that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter. ...
diffusion (disambiguation). ...
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 simulation of airflow into a duct using the Navier-Stokes equations A differential equation is a mathematical equation for an unknown function of one or several variables which relates the values of the function itself and of its derivatives of various orders. ...
## 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. Linear is similarly used to describe regions of any function, mathematical or physical, that follow a straight line with arbitrary slope. This article is about the engineering discipline. ...
For other uses, see Transistor (disambiguation). ...
For the British rock band of the same name, see Amplifier (band). ...
High Fidelity is also the title of a book by Nick Hornby and a film directed by Stephen Frears, based upon Hornbys book. ...
## 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 A formation is a high-level military organization, such as a Brigade, Division, Corps, Army or Army group. ...
A modern recreation of a mid-17th century company of pikemen. ...
It has been suggested that this article or section be merged into Battle of Balaclava. ...
## 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 (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.^{[1]} Art history usually refers to the history of the visual arts. ...
Heinrich WÃ¶lfflin (June 21, 1864 â€“ July 19, 1945) was a famous Swiss art critic, whose objective classifying principles (painterly vs. ...
Renaissance Classicism was a form of art that removed extraneous detail and showed the world as it was. ...
For other uses, see Baroque (disambiguation). ...
â€œDa Vinciâ€ redirects here. ...
This article is about the Renaissance artist. ...
Albrecht DÃ¼rer (pronounced /al. ...
Painterly is a literal translation of German MÃ¤lerisch, hence malerisch, one of the opposed categories popularized by the art historian Heinrich WÃ¶lfflin (1864 - 1945) in order to help focus, enrich and standardize the terms being used by art historians of his time to characterize works of art. ...
Peter Paul Rubens (June 28, 1577 â€“ May 30, 1640) was a prolific seventeenth-century Flemish and European painter, and a proponent of an exuberant Baroque style that emphasized movement, color, and sensuality. ...
Rembrandt Harmenszoon van Rijn (July 15, 1606 â€“ October 4, 1669) was a Dutch painter and etcher. ...
Las Meninas, painted in 1656. ...
It has been suggested that this article be split into multiple articles accessible from a disambiguation page. ...
## Music In music the **linear** aspect is succession, either intervals or melody, as opposed to simultaneity or the vertical aspect. For other uses, see Music (disambiguation). ...
Succession is the act or process of pooing or of following in order or sequence. ...
In music theory, the term interval describes the difference in pitch between two notes. ...
Look up Melody in Wiktionary, the free dictionary In music, a melody is a series of linear events or a succession, not a simultaneity as in a chord. ...
Simultaneity is the property of two events happening at the same time in at least ONE Reference frame. ...
In music theory, the term interval describes the difference in pitch between two notes. ...
## 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". For other uses, see Latin (disambiguation). ...
## References **^** 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 |