Linear maps

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 polynomials

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:

${\displaystyle y=mx+b\ }$

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 b ≠ 0, 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.

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.

Integral linearity

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 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.