Linearity

In mathematics, a linear map or linear function f(x) is a function that satisfies the two properties:[1]

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

These properties are known as the superposition principle. In this definition, x is not necessarily a real number, but can in general be an element of any vector space. A more special definition of linear function, not coinciding with the definition of linear map, is used in elementary mathematics (see below).

Additivity alone implies homogeneity for rational α, since ${\displaystyle f(x+x)=f(x)+f(x)}$ implies ${\displaystyle f(nx)=nf(x)}$ for any natural number n by mathematical induction, and then ${\displaystyle nf(x)=f(nx)=f(m{\tfrac {n}{m}}x)=mf({\tfrac {n}{m}}x)}$ implies ${\displaystyle f({\tfrac {n}{m}}x)={\tfrac {n}{m}}f(x)}$. The density of the rational numbers in the reals implies that any additive continuous function is homogeneous for any real number α, and is therefore linear.

The concept of linearity can be extended to linear operators. Important examples of linear operators include the derivative considered as a differential operator, and other operators constructed from it, such as del and the Laplacian. When a differential 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.

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.

For a description of linear and nonlinear equations, see linear equation.

Boolean functions

Hasse diagram of a linear Boolean function

In Boolean algebra, a linear function is a function ${\displaystyle f}$ for which there exist ${\displaystyle a_{0},a_{1},\ldots ,a_{n}\in \{0,1\}}$ such that

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

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's truth table:

1. 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, 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 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, 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.

In physics, linearity is a property of the differential equations governing many systems; for instance, the Maxwell equations or the diffusion equation.[3]

Linearity of a homogenous 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 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.

In electronics, the linear operating region of a device, for example a transistor, is where an output dependent variable (such as the transistor collector current) is directly proportional 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 a high fidelity audio amplifier, which must amplify a signal without changing its waveform. Others are linear filters, 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.[4]

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.

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.[7] 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.

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

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

• The dictionary definition of linearity at Wiktionary