linear

• adj.

• 直線的な、直鎖状の、リニアな、(数学)線形の、比例した

関

linearly、normal、proportion、proportional、proportionally、proportionate、proportionately、straight

　 　 　 　 　

WordNet

1. measured lengthwise; "cost of lumber per running foot" (同)running
2. designating or involving an equation whose terms are of the first degree (同)additive
3. (of a leaf shape) long and narrow (同)elongate
4. of or in or along or relating to a line; involving a single dimension; "a linear measurement" (同)one-dimensional
5. give pleasant proportions to; "harmonize a building with those surrounding it"
6. the relation between things (or parts of things) with respect to their comparative quantity, magnitude, or degree; "an inordinate proportion of the book is given over to quotations"; "a dry martini has a large proportion of gin" (同)ratio
7. adjust in size relative to other things
8. the quotient obtained when the magnitude of a part is divided by the magnitude of the whole
9. magnitude or extent; "a building of vast proportions" (同)dimension
10. harmonious arrangement or relation of parts or elements within a whole (as in a design); "in all perfectly beautiful objects there is found the opposition of one part to another and a reciprocal balance"- John Ruskin (同)proportionality, balance
11. not homosexual
12. a poker hand with 5 consecutive cards (regardless of suit)
13. successive (without a break); "sick for five straight days" (同)consecutive
14. characterized by honesty and fairness; "straight dealing"; "a square deal" (同)square
15. erect in posture; "sit straight"; "stood defiantly with unbowed back" (同)unbent, unbowed
16. (of hair) having no waves or curls; "her naturally straight hair hung long and silky"
17. following a correct or logical method; "straight reasoning"
18. free from curves or angles; "a straight line"
19. having no deviations; "straight lines"; "straight roads across the desert"; "straight teeth"; "straight shoulders"
20. in a straight line; in a direct course; "the road runs straight"
21. in keeping with the facts; "set the record straight"; "made sure the facts were straight in the report"
22. neatly arranged; not disorderly; "the room is straight now"
23. in accordance with scientific laws
24. being approximately average or within certain limits in e.g. intelligence and development; "a perfectly normal child"; "of normal intelligence"; "the most normal person Ive ever met"
25. conforming with or constituting a norm or standard or level or type or social norm; not abnormal; "serve wine at normal room temperature"; "normal diplomatic relations"; "normal working hours"; "normal word order"; "normal curiosity"; "the normal course of events"
26. forming a right angle
27. having a constant ratio
28. one of the quantities in a mathematical proportion
29. properly related in size or degree or other measurable characteristics; usually followed by `to; "the punishment ought to be proportional to the crime"; "earnings relative to production" (同)relative
30. in proportion; "the height of this wall must be reduced proportionately to give the room pleasant dimensions"
31. in proportion (同)pro_rata
32. to a proportionate degree; "your salary will rise proportionately to your workload" (同)proportionally
33. being in due proportion; "proportionate representation of a minority group"
34. agreeing in amount, magnitude, or degree
35. a long slender leaf (同)elongate leaf
36. mark with lines; "sorrow had lined his face"
37. a length (straight or curved) without breadth or thickness; the trace of a moving point
38. a single frequency (or very narrow band) of radiation in a spectrum
39. reinforce with fabric; "lined books are more enduring"
40. a particular kind of product or merchandise; "a nice line of shoes" (同)product line, line of products, line of merchandise, business line, line of business
41. the road consisting of railroad track and roadbed (同)railway line, rail line
42. a commercial organization serving as a common carrier
43. acting in conformity; "in line with"; "he got out of line"; "toe the line"
44. something (as a cord or rope) that is long and thin and flexible; "a washing line"
45. a conceptual separation or distinction; "there is a narrow line between sanity and insanity" (同)dividing_line, demarcation, contrast
46. a formation of people or things one behind another; "the line stretched clear around the corner"; "you must wait in a long line at the checkout counter"
47. a formation of people or things one beside another; "the line of soldiers advanced with their bayonets fixed"; "they were arrayed in line of battle"; "the cast stood in line for the curtain call"
48. a fortified position (especially one marking the most forward position of troops); "they attacked the enemys line"
49. a mark that is long relative to its width; "He drew a line on the chart"
50. a spatial location defined by a real or imaginary unidimensional extent
51. in games or sports; a mark indicating positions or bounds of the playing area
52. persuasive but insincere talk that is usually intended to deceive or impress; "`let me show you my etchings is a rather worn line"; "he has a smooth line but I didnt fall for it"; "that salesman must have practiced his fast line of talk"
53. text consisting of a row of words written across a page or computer screen; "the letter consisted of three short lines"; "there are six lines in every stanza"
54. be in line with; form a line along; "trees line the riverbank" (同)run along
55. cover the interior of; "line the gloves"; "line a chimney"
56. fill plentifully; "line ones pockets"

PrepTutorEJDIC

1. 線の,直線の / 線から成る,線を使った;線状の / (方程式が)一次の
2. 《the~》(…に対する…の)『割合』,比率《+『of』+『名』+『to』+『名』》 / 〈U〉(…との)『調和』,均衡《+『to』+『名』》 / 〈C〉部分;割り前,分け前 / 《複数形で》大きさ / 〈U〉比例 / …‘の'調和をとる;(…に)…‘を'釣り合わせる《+『名』+『to』+『名』》
3. 『まっすぐな』,一直線の / (毛髪などが)まっすぐな / (背などが,曲らずに)『直立した』,まっすぐな / 《補語にのみ用いて》『整理した』,きちんとした / 真実を言う,率直な,正直な / 連続した / 《米》(ある政党に)徹底した,ベッタリの,きっすいの / (比較変化なし)『まっすぐに』,一直線に;『直立して』,垂直に / 『直ちに』,回り道をしないで / 率直に,端的に / 《the~》まっすぐ,一直線;直線コース / (カードゲームで)ポーカーの5枚続き,ストレート
4. 『正常な』,『通常の』,標準の / (人が肉体・知能・情緒の発達において)平均的な,年齢相応の / (線などが)垂直な;(…に)垂直な《+『to』+『名』》・普通の;典型的な;直角をなす (to)・（実験動物が）常態の, 未処置［未免疫］の;（濃度が）規定の・常態;垂線, 法線;平均量［値］正常,通常;標準
5. 比例の,比例による;(…と)比例として《+『to』+『名』》 / 釣り合った,調和した;(…に)釣り合った《+『to』+『名』》
6. 釣り合いのとれた,比例した;(…と)釣り合いのとれた《+『to』+『名』》
7. 『線』,筋,けい / (色・縫い目などの)『線』,しま;(顔・手などの)しわ(wrinkle) / (物・人の)『列』,行列(row) / 『綱』,糸,針金 / 電線,電話線,電信線 / (文章の)『行』;(詩の)『行』 / 《しばしばa~》《話》『短い手紙』(note) / 《複数形で》輪郭,外形(outline) / 境界線(boundary);限界,限度(limit) / (鉄道・バスなどの)路線;航路,空路 / 《複数形で》せりふ / 進路,道順(course);(考えなどの)筋道,方針 / 《しばしばone's~》(活動の)分野,方面;特意 / 家系,血統,血筋 / 在庫商品,仕入れ品 / 導管,パイプライン / 《しばしば複数形で》戦線,防御線 / …‘に'『線を引く』,筋(しま,しわ)をつける / …‘に'『沿って並ぶ;』(…を)…‘に'沿って並べる《+『名』+『with』+『名』》
8. (材料で)〈着物など〉‘に'裏を付ける《+『名』『with』+『名』》 / (…で)…‘を'満たす,(…を)…‘に'詰め込む《+『名』+『with』+『名』》

Wikipedia preview

出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2017/02/27 20:45:43」(JST)

wiki en

[Wiki en表示]

"Linear" redirects here. For other uses, see Linear (disambiguation).

Not to be confused with Lineage (disambiguation).

Linearity is the property of a mathematical relationship or function which means that it can be graphically represented as a straight line. Examples are the relationship of voltage and current across a resistor (Ohm's law), or the mass and weight of an object. Proportionality implies linearity, but linearity does not imply proportionality

In mathematics

In mathematics, a linear map or linear function f(x) is a function that satisfies the following two properties:

• Additivity: f(x + y) = f(x) + f(y).
• Homogeneity of degree 1: f(αx) = αf(x) for all α.

The homogeneity and additivity properties together are called the superposition principle. It can be shown that additivity implies homogeneity in all cases where α is rational; this is done by proving the case where α is a natural number by mathematical induction and then extending the result to arbitrary rational numbers. If f is assumed to be continuous as well, then this can be extended to show homogeneity for any real number α, using the fact that rationals form a dense subset of the reals.

In this definition, x is not necessarily a real number, but can in general be a member of any vector space. A more specific definition of linear function, not coinciding with the definition of linear map, is used in elementary mathematics.

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 generally straightforward to solve by breaking the equation up into smaller pieces, solving each of those pieces, and summing the solutions.

Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations (also called linear maps), and systems of linear equations.

The word linear comes from the Latin word linearis, which means pertaining to or resembling a line. For a description of linear and nonlinear equations, see linear equation. Nonlinear equations and functions are of interest to physicists and mathematicians because they can be used to represent many natural phenomena, including chaos.

Linear polynomials

Main article: linear equation

In a different usage to the above definition, 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 equation is one of the forms:

$f(x)=mx+b\$

where 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 a function $f$ for which there exist $a_{0},a_{1},\ldots ,a_{n}\in \{0,1\}$ such that

$f(b_{1},\ldots ,b_{n})=a_{0}\oplus (a_{1}\land b_{1})\oplus \cdots \oplus (a_{n}\land b_{n})$, where $b_{1},\ldots ,b_{n}\in \{0,1\}.$

A Boolean function is linear if one of the following holds for the function's truth table:

1. 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. Specifically, f('F', 'F', ..., 'F') = 'F', and these functions correspond to linear maps over the Boolean vector space.
2. In every row in which the value of the function is 'T', there is 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. In this case, f('F', 'F', ..., 'F') = 'T'.

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 functions.

Physics

In physics, linearity is a property of the differential equations governing many systems; for instance, the Maxwell equations or the diffusion equation.

Linearity of a differential equation means that if two functions f and g are solutions of the equation, then any linear combination af + bg is, too.

In instrumentation, linearity means that for every change in the variable you are observing, you get 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 useful certain range, and most useful within that range. In contrast, human senses are highly nonlinear- for instance, the brain totally ignores incoming light unless it exceeds a certain absolute threshold number of photons.

Electronics

In electronics, the linear operating region of a device, for example a transistor, is where a dependent variable (such as the transistor collector current) is directly proportional to an independent 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 high fidelity audio amplifier, which must amplify a signal without changing its waveform. Others are linear filters, linear regulators, and linear amplifiers in general.

In most scientific and technological, 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, taking it away from the approximately linear part of the transfer function.

Integral linearity

Main article: Integral linearity

For an electronic device (or other physical device) that converts a quantity to another quantity, Bertram S. Kolts writes:

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.

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 century (Peter Paul Rubens, Rembrandt, and Velázquez) because they primarily use outline to create shape. Linearity in art can also be referenced in digital art. For example, hypertext fiction can be an example of nonlinear narrative, but there are also websites designed to go in a specified, organized manner, following a linear path.

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.[1] The words "linear"[2] & "lineal" [3] both descend from the same root meaning, the Latin word for line, which is "linea".

See also

• Linear actuator
• Linear element
• Linear system
• Linear medium
• Linear programming
• Linear differential equation
• Bilinear
• Multilinear
• Linear motor
• Linear A and Linear B scripts.
• Linear interpolation

References

External links

• The dictionary definition of linearity at Wiktionary

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Related Pictures

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★リンクテーブル★

リンク元	「normal」「比例」「proportional」「直鎖状」「proportionally」
拡張検索	「nonlinear」「non-linear」「nonlinearity」
関連記事	「line」

「normal」

　　[★]

• adj.

• 正常な、正常型の、普通の、通常の、(統計)正規の、(化学)規定濃度、直鎖状の、(直鎖状の)ノルマル

関

common、commonly、conventional、general、generally、in general、linear、N、n-、normally、ordinarily、ordinary、straight、usual、usually

　 　 　 　 　

「比例」

　　[★]

英

proportion、proportional、proportionate、linear、proportionally、proportionately

関

線形、直鎖状、比率、比例的、リニア、割合、直線的、釣り合い、釣り合った

「proportional」

　　[★]

• adj.

• 比例した(to)、釣り合った。比例の

関

balanced、commensurable、commensurate、linear、proportion、proportionally、proportionate、proportionately、rate

　 　 　 　

「直鎖状」

　　[★]

英

linear、straight、normal

関

正常、線形、直線、通常、ノルマル、比例、普通、リニア、直線的、規定濃度、正規、正常型

「proportionally」

　　[★]

• adv.

• 比例的に、比例して

関

linear、proportion、proportional、proportionate、proportionately

「nonlinear」

　　[★]

• adj.

• (数学)非線形の、非直線性の

関

non-linear、nonlinearity

　

「non-linear」

　　[★]

• 非線形の

関

nonlinear

「nonlinearity」

　　[★]

• 非直線性

関

nonlinear

「line」

　　[★]

• n.

• 線、ライン、系統

• v.

• 裏打ちする

関

lineage、lining、pedigree、stock、strain、string、undercoat