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ID: 1813530 • Letter: #
Question
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Explanation / Answer
Linearity
"Linear" redirects here. For other uses, see Linear (disambiguation).
Not to be confused with Lineage (disambiguation).
In mathematics, a linear map or linear function f(x) is a function which satisfies the following two properties:
Additivity (also called the superposition property): f(x + y) = f(x) + f(y).
Homogeneity of degree 1: f(?x) = ?f(x) for all ?.
It can be shown that additivity implies the homogeneity in all cases where ? is rational; this is done by proving the case where ? 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 that homogeneity for ? any real number, 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 particularly easy 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.
Integral linearity[edit source
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.
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.
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[edit source
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) = m x + 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[edit source
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) for all 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:
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.
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[edit source | editbeta]
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.
Electronics[edit source | editbeta]
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
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