# Dictionary Definition

linear adj

1 designating or involving an equation whose
terms are of the first degree [syn: additive] [ant: nonlinear]

2 of or in or along or relating to a line;
involving a single dimension; "a linear foot" [syn: one-dimensional]
[ant: planar, cubic]

3 of a circuit or device having an output that is
proportional to the input; "analogue device"; "linear amplifier"
[syn: analogue,
analog] [ant: digital]

4 of a leaf shape; long and narrow [syn: elongate]

5 measured lengthwise; "cost of lumber per
running foot" [syn: running(a)]

# User Contributed Dictionary

## English

### Pronunciation

- /ˈlɪ.ni.ɚ/

### Adjective

- Having the form of a line; straight.
- Pertaining or related to lines.
- Made in a step-by-step, logical manner.
- botany of leaves Long and narrow, with nearly parallel sides.
- Of, or related to a class of polynomial of the form y = a.x + b .
- A type of length measurement involving only one spatial dimension ib as opposed to area or volume.

#### Translations

in mathematics, of polynomial

- Czech: lineární

### Related terms

# Extensive Definition

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:

- 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(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.

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.

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.

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.

## 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.

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.

## 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.

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.

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).

## Boolean functions

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

If there exists a_0, a_1, \ldots, a_n \in \ such
that f(b_1, \ldots, b_n) = a_0 \oplus (a_1 \land b_1) \oplus \ldots
\oplus (a_n \land b_n), \forall b_1, \ldots , b_n \in \

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.

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.

## 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).

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.

## 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.

## 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.

## 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.## Music

In music
the linear aspect is succession, either intervals
or melody, as opposed to
simultaneity or the
vertical
aspect.

## 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. http://www.unc.edu/~rowlett/units/dictL.html The words "linear"http://www.yourdictionary.com/ahd/l/l0180100.html & "lineal" http://www.yourdictionary.com/ahd/l/l0180300.html both descend from the same root meaning, the Latin word for line, which is "linea".## References

## See also

linear in Danish: Lineær

linear in German: Linear

linear in Esperanto: Vikipedio:Projekto
matematiko/Lineara

linear in Korean: 선형성

linear in Japanese: 線型性

linear in Dutch: Lineariteit

linear in Norwegian: Lineær

linear in Slovenian: Linearna funkcija

linear in Swedish: Linjär funktion

linear in Chinese: 線性關係

'''# Synonyms, Antonyms and Related Words

Plimsoll line, Plimsoll mark, arrowlike, catenary, consecutive, dead straight,
direct, even, flat, high-water mark, horizontal, in a line,
level, lineal, load waterline, ordinal, progressive, rectilineal, rectilinear, right, ruler-straight, sequent, sequential, serial, seriate, smooth, straight, straight-cut,
straight-front, straight-side, streamlined, successional, successive, tidemark, true, unbending, unbent, unbowed, unbroken, uncurved, undeflected, undeviating, undistorted, uninterrupted, unswerving, unturned, upright, vertical, waterline, watermark