Contents 1 In mathematics 1.1 Linear polynomials 1.2 Boolean functions 2 Physics 3 Electronics 3.1 Integral linearity 4 Military tactical formations 5 Art 6 Music 7 Measurement 8 See also 9 References 10 External links In mathematics[edit] In mathematics, a linear map or linear function f(x) is a function that satisfies the following two properties:[1] Additivity: f(x + y) = f(x) + f(y). Homogeneity of degree 1: f(αx) = αf(x) for all α. The homogeneity and additivity properties together are called the
superposition principle. It can be shown that additivity implies
homogeneity in all cases where α is rational; this is done by proving
the case where α is a natural number by mathematical induction and
then extending the result to arbitrary rational numbers. If f is
assumed to be continuous as well, then this can be extended to show
homogeneity for any real number α, using the fact that rationals form
a dense subset of the reals.
In this definition, x is not necessarily a real number, but can in
general be a member of any vector space. A more specific definition of
linear function, not coinciding with the definition of linear map, is
used in elementary mathematics.
The concept of linearity can be extended to linear operators.
Important examples of linear operators include the derivative
considered as a differential operator, and many constructed from it,
such as del and the Laplacian. When a differential equation can be
expressed in linear form, it is generally straightforward to solve by
breaking the equation up into smaller pieces, solving each of those
pieces, and summing the solutions.
f ( x ) = m x + b displaystyle f(x)=mx+b 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 b = 0. Hence, if b ≠ 0, the function is often called an affine function (see in greater generality affine transformation). Boolean functions[edit] In Boolean algebra, a linear function is a function f displaystyle f for which there exist a 0 , a 1 , … , a n ∈ 0 , 1 displaystyle a_ 0 ,a_ 1 ,ldots ,a_ n in 0,1 such that f ( b 1 , … , b n ) = a 0 ⊕ ( a 1 ∧ b 1 ) ⊕ ⋯ ⊕ ( a n ∧ b n ) displaystyle 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 , … , b n ∈ 0 , 1 . displaystyle b_ 1 ,ldots ,b_ n in 0,1 . 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, f(F, F, ..., F) = F, 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, 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.
Physics[edit]
In physics, linearity is a property of the differential equations
governing many systems; for instance, the
This section contains too-lengthy quotations for an encyclopedic entry. Please help improve the article by presenting facts as a neutrally-worded summary with appropriate citations. Consider transferring direct quotations to Wikiquote. (September 2014) 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.
This section needs expansion. You can help by adding to it. (March 2013) See also[edit] Linear actuator
Linear element
Linear system
Linear medium
Linear programming
Linear differential equation
Bilinear
Multilinear
Linear motor
References[edit] ^ Edwards, Harold M. (1995). Linear Algebra. Springer. p. 78.
ISBN 9780817637316.
^ Stewart, James (2008). Calculus: Early Transcendentals, 6th ed.,
Brooks Cole Cengage Learning. ISBN 978-0-495-01166-8, Section 1.2
^ Evans, Lawrence C. (2010) [1998], Partial differential equations
(PDF), Graduate Studies in Mathematics, 19 (2nd ed.), Providence,
R.I.: American Mathematical Society, ISBN 978-0-8218-4974-3,
MR 2597943
^ Whitaker, Jerry C. (2002). The RF transmission systems handbook. CRC
Press. ISBN 978-0-8493-0973-1.
^ Kolts, Bertram S. (2005). "Understanding
External links[edit] The dictionary definition of linea |