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 straight line. Over the reals, a linear equation is one of the forms: 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 in the section 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. Hence, if, the function is often called an affine function.
Boolean functions
In Boolean algebra, a linear function is a function for which there exist such that Note that if, the above function is considered affine in linear algebra. 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 Ts assigned to the arguments, and in every row in which the function is F there is an even number of Ts assigned to arguments. Specifically,, 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 Ts 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 Ts assigned to arguments. In this case,.
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 homogenous differential equation means that if two functions f and g are solutions of the equation, then any linear combination 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 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 is directly proportional to an independent variable. This ensures that an analog output is an accurate representation of an input, typically with higher amplitude. 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 ; 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
For an electronic 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 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.
Military tactical formations
In military tactical formations, "linear formations" were adapted starting from phalanx-like formations of pike protected by handgunners, towards shallow formations of handgunners protected by progressively fewer pikes. This kind of formation got progressively thinner until its extreme in the age of Wellington's 'Thin Red Line'. It was eventually replaced by skirmish order when the invention of the breech-loading rifle allowed soldiers to move and fire in small, mobile units, unsupported by large-scale formations of any shape.
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 are more linear than "painterly" Baroque painters of the seventeenth century 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 generally without regard to the width. It is sometimes incorrectly referred to as "lineal feet"; however, "lineal" is typically used to refer to lines of ancestry or heredity.
