In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, and more specifically in
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrice ...
, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a
mapping between two
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s that preserves the operations of
vector addition and
scalar multiplication. The same names and the same definition are also used for the more general case of
modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
over a
ring; see
Module homomorphism.
If a linear map is a
bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
then it is called a . In the case where
, a linear map is called a (linear) ''
endomorphism''. Sometimes the term refers to this case, but the term "linear operator" can have different meanings for different conventions: for example, it can be used to emphasize that
and
are
real vector spaces (not necessarily with
), or it can be used to emphasize that
is a
function space
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vect ...
, which is a common convention in
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defi ...
. Sometimes the term ''
linear function'' has the same meaning as ''linear map'', while in
analysis
Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (3 ...
it does not.
A linear map from ''V'' to ''W'' always maps the origin of ''V'' to the origin of ''W''. Moreover, it maps
linear subspace
In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, l ...
s in ''V'' onto linear subspaces in ''W'' (possibly of a lower
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
); for example, it maps a
plane through the
origin
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* The Origin (Buffy comic), ''The Origin'' (Bu ...
in ''V'' to either a plane through the origin in ''W'', a
line
Line most often refers to:
* Line (geometry), object with zero thickness and curvature that stretches to infinity
* Telephone line, a single-user circuit on a telephone communication system
Line, lines, The Line, or LINE may also refer to:
Art ...
through the origin in ''W'', or just the origin in ''W''. Linear maps can often be represented as
matrices, and simple examples include
rotation and reflection linear transformations.
In the language of
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
, linear maps are the
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
s of vector spaces.
Definition and first consequences
Let
and
be vector spaces over the same
field .
A function
is said to be a ''linear map'' if for any two vectors
and any scalar
the following two conditions are satisfied:
*
Additivity / operation of addition
*
Homogeneity
Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, size, ...
of degree 1 / operation of scalar multiplication
Thus, a linear map is said to be ''operation preserving''. In other words, it does not matter whether the linear map is applied before (the right hand sides of the above examples) or after (the left hand sides of the examples) the operations of addition and scalar multiplication.
By
the associativity of the addition operation denoted as +, for any vectors
and scalars
the following equality holds:
Thus a linear map is one which preserves
linear combinations.
Denoting the zero elements of the vector spaces
and
by
and
respectively, it follows that
Let
and
in the equation for homogeneity of degree 1:
A linear map
with
viewed as a one-dimensional vector space over itself is called a
linear functional.
These statements generalize to any left-module
over a ring
without modification, and to any right-module upon reversing of the scalar multiplication.
Examples
* A prototypical example that gives linear maps their name is a function
, of which the
graph is a line through the origin.
* More generally, any
homothety where
centered in the origin of a vector space is a linear map.
* The zero map
between two vector spaces (over the same
field) is linear.
* The
identity map
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unc ...
on any module is a linear operator.
* For real numbers, the map
is not linear.
* For real numbers, the map
is not linear (but is an
affine transformation
In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.
More generall ...
).
* If
is a
real matrix, then
defines a linear map from
to
by sending a
column vector to the column vector
. Conversely, any linear map between
finite-dimensional
In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to d ...
vector spaces can be represented in this manner; see the , below.
* If
is an
isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' ...
between real
normed space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length ...
s such that
then
is a linear map. This result is not necessarily true for complex normed space.
*
Differentiation defines a linear map from the space of all differentiable functions to the space of all functions. It also defines a linear operator on the space of all
smooth function
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
s (a linear operator is a linear
endomorphism, that is, a linear map with the same
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
** Natural domain of a partial function
**Domain of holomorphy of a function
* ...
and
codomain). An example is
* A definite
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
over some
interval is a linear map from the space of all real-valued integrable functions on to
. For example,
* An indefinite
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
(or
antiderivative
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolica ...
) with a fixed integration starting point defines a linear map from the space of all real-valued integrable functions on
to the space of all real-valued, differentiable functions on
. Without a fixed starting point, the antiderivative maps to the
quotient space of the differentiable functions by the linear space of constant functions.
* If
and
are finite-dimensional vector spaces over a field , of respective dimensions and , then the function that maps linear maps
to matrices in the way described in (below) is a linear map, and even a
linear isomorphism
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
.
* The
expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
of a
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
(which is in fact a function, and as such a element of a vector space) is linear, as for random variables
and
we have