In mathematics, the term linear is used in two distinct senses for two different properties:
An example of a linear function is the function defined by
f(x)=(ax,bx)
X,
Y
Z
aX+bY+cZ+d.
Linearity of a mapping is closely related to proportionality. Examples in physics include the linear relationship of voltage and current in an electrical conductor (Ohm's law), and the relationship of mass and weight. By contrast, more complicated relationships, such as between velocity and kinetic energy, are nonlinear.
Generalized for functions in more than one dimension, linearity means the property of a function of being compatible with addition and scaling, also known as the superposition principle.
Linearity of a polynomial means that its degree is less than two. The use of the term for polynomials stems from the fact that the graph of a polynomial in one variable is a straight line. In the term "linear equation", the word refers to the linearity of the polynomials involved.
Because a function such as
f(x)=ax+b
The word linear comes from Latin linearis, "pertaining to or resembling a line".
In mathematics, a linear map or linear function f(x) is a function that satisfies the two properties:[1]
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
f(x+x)=f(x)+f(x)
f(nx)=nf(x)
nf(x)=f(nx)=f(m\tfrac{n}{m}x)=mf(\tfrac{n}{m}x)
f(\tfrac{n}{m}x)=\tfrac{n}{m}f(x)
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.
See main article: Linear equation and Linear algebra. 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.[2]
Over the reals, a simple example of a linear equation is given by:
y=mx+b
where m is often called the slope or gradient, and 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 the constant term – b in the example – equals 0. If, the function is called an affine function (see in greater generality affine transformation).
Linear algebra is the branch of mathematics concerned with systems of linear equations.
See main article: article and Parity function. In Boolean algebra, a linear function is a function
f
a0,a1,\ldots,an\in\{0,1\}
f(b1,\ldots,bn)=a0 ⊕ (a1\landb1) ⊕ … ⊕ (an\landbn)
b1,\ldots,bn\in\{0,1\}.
Note that if
a0=1
A Boolean function is linear if one of the following holds for the function's truth table:
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.
See main article: Superposition principle. 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.
Linear motion traces a straight line trajectory.
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.[3]
See main article: Integral linearity. For an electronic device (or other physical device) that converts a quantity to another quantity, Bertram S. Kolts writes:[4] [5]
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.