7.9 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Linearity | 1/2 | https://en.wikipedia.org/wiki/Linearity | reference | science, encyclopedia | 2026-05-05T07:24:16.618385+00:00 | kb-cron |
In mathematics, the term linear is used in two distinct senses for two different properties:
linearity of a function (or mapping); linearity of a polynomial. An example of a linear function is the function defined by
f
(
x
)
=
(
a
x
,
b
x
)
{\displaystyle f(x)=(ax,bx)}
that maps the real line to a line in the Euclidean plane R2 that passes through the origin. An example of a linear polynomial in the variables
X
,
{\displaystyle X,}
Y
{\displaystyle Y}
and
Z
{\displaystyle Z}
is
a
X
+
b
Y
+
c
Z
+
d
.
{\displaystyle 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
)
=
a
x
+
b
{\displaystyle f(x)=ax+b}
is defined by a linear polynomial in its argument, it is sometimes also referred to as being a "linear function", and the relationship between the argument and the function value may be referred to as a "linear relationship". This is potentially confusing, but usually the intended meaning will be clear from the context. The word linear comes from Latin linearis, "pertaining to or resembling a line".
== In mathematics ==
=== Linear maps === In mathematics, a linear map or linear function f(x) is a function that satisfies the two properties:
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
f
(
x
+
x
)
=
f
(
x
)
+
f
(
x
)
{\displaystyle f(x+x)=f(x)+f(x)}
implies
f
(
n
x
)
=
n
f
(
x
)
{\displaystyle f(nx)=nf(x)}
for any natural number n by mathematical induction, and then
n
f
(
x
)
=
f
(
n
x
)
=
f
(
m
n
m
x
)
=
m
f
(
n
m
x
)
{\displaystyle nf(x)=f(nx)=f(m{\tfrac {n}{m}}x)=mf({\tfrac {n}{m}}x)}
implies
f
(
n
m
x
)
=
n
m
f
(
x
)
{\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. Over the reals, a simple example of a linear equation is given by
y
=
m
x
+
b
,
{\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 ===
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\}.}
Note that if
a
0
=
1
{\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: