Linear maps In mathematics, a
linear map or
linear function f(
x) is a function that satisfies the two properties: •
Additivity: . •
Homogeneity of degree 1: 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 f(x+x)=f(x)+f(x) implies f(nx)=n f(x) for any
natural number n by
mathematical induction, and then n f(x) = f(nx)=f(m\tfrac{n}{m}x)= m f(\tfrac{n}{m}x) implies 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. Over the reals, a simple example of a
linear equation is given by y = m x + 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.
Boolean functions of a linear Boolean function 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), where b_1, \ldots, b_n \in \{0,1\}. Note that if 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: • 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, . 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==