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 between two
vector spaces that preserves the operations of
vector addition
In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors ...
and
scalar multiplication. The same names and the same definition are also used for the more general case of
modules over a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
; 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
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
vector spaces (not necessarily with
), or it can be used to emphasize that
is a
function space, which is a common convention in
functional analysis. Sometimes the term ''
linear function
In mathematics, the term linear function refers to two distinct but related notions:
* In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. For dist ...
'' 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 subspaces in ''V'' onto linear subspaces in ''W'' (possibly of a lower
dimension); for example, it maps a
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
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, cate ...
, 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
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
.
A function
is said to be a ''linear map'' if for any two vectors
and any scalar
the following two conditions are satisfied:
*
Additivity
Additive may refer to:
Mathematics
* Additive function, a function in number theory
* Additive map, a function that preserves the addition operation
* Additive set-functionn see Sigma additivity
* Additive category, a preadditive category with fi ...
/ 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
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If is a vector space over a field , the s ...
.
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
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
) is linear.
* The
identity map 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 generally, ...
).
* If
is a
real matrix, then
defines a linear map from
to
by sending a
column vector
In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example,
\boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end.
Similarly, a row vector is a 1 \times n matrix for some n, c ...
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 dist ...
vector spaces can be represented in this manner; see the , below.
* If
is an
isometry 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 functions (a linear operator is a linear
endomorphism, that is, a linear map with the same
domain and
codomain
In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either ...
). An example is
* A definite
integral over some
interval is a linear map from the space of all real-valued integrable functions on to
. For example,
* An indefinite
integral (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 symbolical ...
) 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.
* 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 (which is in fact a function, and as such a element of a vector space) is linear, as for random variables
and
we have