Linearity

Linearity ( Latin linea "line", linearis "consisting of lines") has a different meaning in different areas, but mostly describes a straight line structure.

Linearity in Science and Technology

general definition

Linearity is the property of a system to always react to a change in one parameter with a proportional change in another parameter.

This general definition applies equally to systems theory, technology, physics and mathematics. If it is not fulfilled, one speaks of non-linearity .

Linearity in Mathematics

Example: linear functions

The easiest way to explain linearity in mathematics is by means of so-called linear functions . These are special functions in which one or more variables are mapped onto another according to the general definition. Linearity is also a property of various mathematical transformations such as B. the Z transformation or the continuous Fourier transformation .

Furthermore, one encounters the concept of linearity (partially tightened there) in the following mathematical topics:

• in linear algebra , the study of vector spaces
• in the case of a linear mapping , a structure-preserving mapping between two vector spaces
• in a linear equation , a special type of equations such as linear ordinary differential equations
• for order relations in the concept of linear order

See also: proportionality (as a special case of linearity)

Linearity in statistical models

The statistics provides methods , with the help of statistical models in linear and non-linear models can be distinguished. The specialty: linear models are linear in the estimated parameters , but not necessarily in the independent variables. What do you mean with that?

Explanation: In the quadratic equation, it is not the independent variable , but the parameters of the model (here the coefficients :) that is decisive for the existence of linearity. Because of this, multiple linear regression can be used to estimate parameters of “curved” models. ${\displaystyle ax^{2}+bx+c=0}$ ${\displaystyle x}$ ${\displaystyle a,b,c}$

See also : collinearity  and multicollinearity

Linearity in method validation

In method validation , as it is e.g. B. is used in analytical chemistry or forensics , a statistical test for linearity according to Mandel, the so-called Mandel test (or Mandel adaptation test) is common, which is used to determine which regression model ( linear or quadratic regression ) for the case at hand is to be assumed.

Linearity in Physics and Technology

Common

The concept of linearity is based on a general linear function between real values ​​of two physical values. Are an input variable and an output variable suitable for describing a component (a device, a device) or a physical relationship , and do these variables satisfy the equation? ${\displaystyle x}$ ${\displaystyle y}$

${\displaystyle y=m\cdot x+b\quad {\text{mit }}\ m,b={\text{reelle Konstanten,}}}$

one speaks of a linear component or a linear relationship. Equivalent to this linear function is the statement that

${\displaystyle {\frac {\Delta y}{\Delta x}}=m={\text{konst}}}$

is - independent of the size of and from the working point or starting point from which and count. This corresponds to the “general definition” given above. ${\displaystyle \Delta x}$ ${\displaystyle \Delta x}$ ${\displaystyle \Delta y}$

In the special case that is, the property is characterized by proportionality . Then also applies ${\displaystyle b=0}$

${\displaystyle {\frac {y}{x}}=m={\text{konst .}}}$

In a right-angled coordinate system with equally divided axes, the linear relationship between the output signal and the input signal is represented by a straight characteristic curve . In the case of a proportional relationship, this goes through the coordinate origin .

In the case of a continuously curved characteristic curve, a linear approximation can be used within the framework of a small-signal behavior , provided that the deviation of the curve from its tangent (in the respectively selected operating point) is still small for small values ​​of . ${\displaystyle |\Delta x|}$

Scales of an analog multimeter (only the middle scale is linearly divided)

Examples from measurement technology

• Often there is a linear function between a measured variable (e.g. concentration of a substance in analytical chemistry) and the measurement signal (e.g. the electrical voltage of a sensor ). The aim for a measuring device is as proportional as possible. For this purpose, not only is the signal amplified in a measuring chain as part of signal processing , but the zero point is also shifted if necessary.

• In the picture on the right, there is a non-linear relationship between the measured variable and the deflection or angle of the pointer in the measuring range belonging to the upper scale . Thanks to a non-linear scale division, the readable value is nevertheless proportional to the measured variable.

See also : linearity (measurement system analysis ) , linearity deviation in the digital multimeter and linearity condition for correlation coefficients

Typical transfer characteristic of a field effect transistor

Examples from electrical engineering

• The fundamental passive components ohmic resistance , coil and capacitor are called linear resistances in an alternating current circuit , since they react to a harmonic oscillation of the input variable with a harmonic oscillation of the output variable at the same frequency . These quantities are electrical voltage and electrical current strength . - Semiconductor components generally behave non-linearly. ${\displaystyle \Delta x}$ ${\displaystyle \Delta y}$ ${\displaystyle U}$ ${\displaystyle I}$

• The relationship between the control voltage and the controlled current intensity of a field effect transistor is shown in the characteristic curve on the right. Two areas can be distinguished, whereby the boundary is fluid. ${\displaystyle U_{\mathrm {GS} }}$ ${\displaystyle I_{\mathrm {D} }}$
  • In the range 0… −1 V the curve can be viewed as a straight line to a good approximation; there is linearity . Here, a voltage change , which is counted from an operating point, is followed by a proportional change in the current strength . If there is a sinusoidal change in time , it also follows sinusoidally. ${\displaystyle \Delta U_{\mathrm {GS} }}$ ${\displaystyle \Delta I_{\mathrm {D} }}$ ${\displaystyle \Delta U_{\mathrm {GS} }}$ ${\displaystyle \Delta I_{\mathrm {D} }}$
  • In the range -1 ... -3 V is the function not linear . This results in distortions : If there is a sinusoidal course of time, then follows with a non-sinusoidal course. ${\displaystyle \Delta U_{\mathrm {GS} }}$ ${\displaystyle \Delta I_{\mathrm {D} }}$

See also : small-signal behavior  and small-signal equivalent circuit diagram

Linearity in Mechanics

According to the Latin meaning of the word linea , u. a. a differentiation of the direction of movement of bodies according to whether the movement takes place along a correspondingly pronounced straight line (linear) or not (non-linear). Example: In an internal combustion engine , the reciprocating piston moves in a straight line (uneven translation ) and the crankshaft connected to it makes a circular movement (steady rotation ).

Linearity in Chemistry

In chemical analyzes , the method used is linear if the measurement signal is directly proportional to the analyte concentration in the sample in a certain concentration range.

Linearity in Systems Theory

In interdisciplinary systems theory , a distinction is made between linear and non-linear systems .

A system is linear if the following two criteria are met:

1. If an input variable leads to an output variable , an input variable of will lead to an output variable of . In other words, the magnitude (order of magnitude) of the input variable is proportional to the magnitude of the output variable of the system. ${\displaystyle x}$ ${\displaystyle X}$ ${\displaystyle 2\cdot x}$ ${\displaystyle 2\cdot X}$
2. If an input variable leads to an output variable and an input variable leads to an output variable , an input variable causes an output variable . In other words, the system treats two simultaneous input variables independently of one another and these do not interact with one another within the system. ${\displaystyle x}$ ${\displaystyle X}$ ${\displaystyle y}$ ${\displaystyle Y}$ ${\displaystyle x+y}$ ${\displaystyle X+Y}$

• Implicit in these criteria is the fact that linear systems do not produce any frequencies at the output that are not already available at the input.
• It should also be noted that these criteria do not contain any statements according to which the output variable would be the same as the input variable, or even similar. For example, the input variable could be an electrical current and the output variable a temperature. In the case of mechanical structures such as B. Machines one could imagine an oscillating force as an input variable and a measurable vibration (movement) as an output variable.

System theory also knows linear, time-invariant, dynamic systems .

See also : linearization

Linearity in Ethnosociology

→ Main article : Linear and collateral kinship  and rules of descent (descent)

The Ethnosociology used the term "linearity" to switch between linear relationship and collateral side relationship to distinguish: Linearity denotes the direct lineage of parents , their parents and so on, as well as the entire own offspring ( straight line ), while Kollateralität the indirect relationship between Siblings of all generations and their descendants (sideline).

The linearity of kinship plays an important role in unilinear rules of ancestry that determine unilinear inheritance:

• Patri- linearity : only via the patriarchal line (46 percent of the 1,300 ethnic groups and indigenous peoples worldwide)
• Matri- linearity : only over the maternal line (13 percent)
• Bi- linearity : double, across both lines, one depending on the social context (4 percent)
• Ambi- linearity : a self-chosen,mixed line adoptedby the mother or father (4 percent)
• Parallel linearity : the mother transfers her line to daughters, the father his to sons (1 percent)

Linearity of texts and media

Linearity of texts , for example, in books is when these in a specific linear order be written and read. Characteristics of linearity in texts can be:

• from side to side, or from sheet to sheet
• front to back, or back to front
• from row to row, or from column to column
• left to right, or right to left
• from top to bottom, or from bottom to top

This linearity of the text is followed by the direction of movement of the author's or reader's eyes.

Modern media such as the Internet allow this linearity to be broken up in certain areas with the help of hypertext by linking texts to one another and thus adding a non-linear component to them. The linearity of individual sections is retained, but the reader determines the reading direction by selecting the content of interest based on the hyperlinks set by the authors and thus moves non-linearly through the existing media content. This non-linearity of hypertext is considered an essential element of hypertext theory .

See also : linearity of a character

Linearity in music

→ Main article : Linearity (music)

In music, linearity describes the relationship between one, two or more voices in relation to the respective rules of the musical composition . The fact that this already begins with unanimity is due to the fact that phrases, so-called (final) clauses, which have already been defined, are adhered to or not.

See also

• Linear technology (drive technology )
• Global linearization method (control theory)
• LINEAR: Lincoln Near Earth Asteroid Research
• Linearity of historical change
• David R. Hawkins: Non-Duality and Non-linearity
• Linear-Antiqua (fonts)

Individual evidence

1. ↑ ^{a b} Entry: Definition of Linearity. In: Azima DLI. USA 2009, accessed on December 22, 2013 (English; definition of linearity in systems).
2. ↑ Hans Lohninger: Linear and non-linear models. In: Basics of Statistics. Own website, October 2012, accessed on December 22, 2013.
3. ↑ Lexicon entry: Linearity: Subject - Mathematics. In: chemgapedia.de. Wiley Information Services GmbH, 2013, accessed on December 22, 2013 (linearity in chemical analysis methods, with a 45-minute learning unit).
4. ^ Stefan Münz: Hypertext (1): Text and linearity. In: Web Competence Blog. March 29, 2007, accessed December 22, 2013.
5. ↑ Non-linearity. In: HistnetWiki. March 21, 2006, accessed December 22, 2013 (Non-linearity in hypertext theory).