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In mathematics, the termlinear is used in two distinct senses for two different properties:

• linearity of afunction (ormapping);
• linearity of apolynomial.

An example of a linear function is the function defined by${\displaystyle f(x)=(ax,bx)}$ that maps the real line to a line in theEuclidean planeR² that passes through the origin. An example of a linear polynomial in the variables${\displaystyle X,}$${\displaystyle Y}$ and${\displaystyle Z}$ is${\displaystyle aX+bY+cZ+d.}$

Linearity of a mapping is closely related toproportionality. Examples inphysics include the linear relationship ofvoltage andcurrent in anelectrical conductor (Ohm's law), and the relationship ofmass andweight. By contrast, more complicated relationships, such as betweenvelocity andkinetic energy, arenonlinear.

Generalized for functions in more than onedimension, linearity means the property of a function of being compatible withaddition andscaling, also known as thesuperposition principle.

Linearity of a polynomial means that itsdegree is less than two. The use of the term for polynomials stems from the fact that thegraph of a polynomial in one variable is a straightline. In the term "linear equation", the word refers to the linearity of the polynomials involved.

Because a function such as${\displaystyle f(x)=ax+b}$ is defined by a linear polynomial in its argument, it is sometimes also referred to as being a "linear function", and the relationship between the argument and the function value may be referred to as a "linear relationship". This is potentially confusing, but usually the intended meaning will be clear from the context.

The wordlinear comes fromLatinlinearis, "pertaining to or resembling a line".

In mathematics, alinear map orlinear functionf(x) is a function that satisfies the two properties:^[1]

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

These properties are known as thesuperposition principle. In this definition,x is not necessarily areal number, but can in general be anelement of anyvector space. A more special definition oflinear function, not coinciding with the definition of linear map, is used in elementary mathematics (see below).

Additivity alone implies homogeneity forrational α, since${\displaystyle f(x+x)=f(x)+f(x)}$ implies${\displaystyle f(nx)=nf(x)}$ for anynatural numbern bymathematical induction, and then${\displaystyle nf(x)=f(nx)=f(m\frac{n}{m}x)=mf(\frac{n}{m}x)}$ implies${\displaystyle f(\frac{n}{m}x)=\frac{n}{m}f(x)}$. Thedensity of the rational numbers in the reals implies that any additivecontinuous function is homogeneous for any real number α, and is therefore linear.

The concept of linearity can be extended to linearoperators. Important examples of linear operators include thederivative considered as adifferential operator, and other operators constructed from it, such asdel and theLaplacian. When adifferential equation can be expressed in linear form, it can generally be solved by breaking the equation up into smaller pieces, solving each of those pieces, and summing the solutions.

In a different usage to the above definition, apolynomial of degree 1 is said to be linear, because thegraph of a function of that form is a straight line.^[2]

Over the reals, a simple example of alinear equation is given by$${\displaystyle y=mx+b,}$$wherem is often called theslope orgradient, andb they-intercept, which gives the point of intersection between the graph of the function and they axis.

Note that this usage of the termlinear is not the same as in the section above, because linear polynomials over the real numbers do not in general satisfy either additivity or homogeneity. In fact, they do soif and only if theconstant term – b in the example – equals 0. Ifb ≠ 0, the function is called anaffine function (see in greater generalityaffine transformation).

Linear algebra is the branch of mathematics concerned with systems of linear equations.

InBoolean algebra, a linear function is a function${\displaystyle f}$ for which there exist${\displaystyle a_0,a_1,\dots ,a_n\in \{0,1\}}$ such that

${\displaystyle f(b_1,\dots ,b_n)=a_0\oplus (a_1\wedge b_1)\oplus \cdots \oplus (a_n\wedge b_n)}$, where${\displaystyle b_1,\dots ,b_n\in \{0,1\}.}$

Note that if${\displaystyle a_0=1}$, the above function is considered affine in linear algebra (i.e. not linear).

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

1. In every row in which the truth value of the function isT, there are an odd number of Ts assigned to the arguments, and in every row in which the function isF there is an even number of Ts assigned to arguments. Specifically,f(F, F, ..., F) = F, and these functions correspond tolinear 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 Ts assigned to the arguments of the function; and in every row in which thetruth value of the function is F, there are an odd number of Ts 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 thetruth value of the operation or it never makes a difference.

Negation,Logical biconditional,exclusive or,tautology, andcontradiction are linear functions.

Inphysics,linearity is a property of thedifferential equations governing many systems; for instance, theMaxwell equations or thediffusion equation.^[3]

Linearity of ahomogeneous differential equation means that if two functionsf andg are solutions of the equation, then anylinear combinationaf +bg is, too.

In instrumentation, linearity means that a given change in an input variable gives 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 certain range, and most useful within that range. In contrast, human senses are highly nonlinear: for instance, the brain completely ignores incoming light unless it exceeds a certainabsolute threshold number of photons.

Linear motion traces a straight line trajectory.

Inelectronics, the linear operating region of a device, for example atransistor, is where an outputdependent variable (such as the transistor collectorcurrent) is directlyproportional to an input dependent 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 ahigh fidelityaudio amplifier, which must amplify a signal without changing its waveform. Others arelinear filters, andlinear amplifiers in general.

In mostscientific andtechnological, 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.^[4]

For an electronic device (or other physical device) that converts a quantity to another quantity, Bertram S. Kolts writes:^[5][6]

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 offull 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.

• Linear actuator
• Linear element
• Linear foot
• Linear system
• Linear programming
• Linear differential equation
• Bilinear
• Multilinear
• Linear motor
• Linear interpolation

• The dictionary definition oflinearity at Wiktionary

• Physical phenomena

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