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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, 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 matrix (mat ...
, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a
mapping Mapping may refer to: * Mapping (cartography), the process of making a map * Mapping (mathematics), a synonym for a mathematical function and its generalizations ** Mapping (logic), a synonym for functional predicate Types of mapping * Animated ...
V \to W between two
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s that preserves the operations of
vector addition In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

vector addition
and
scalar multiplication 250px, The scalar multiplications −a and 2a of a vector a In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ...

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 syst ...
over a ring; see
Module homomorphism In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
. If a linear map is a
bijection In , a bijection, bijective function, one-to-one correspondence, or invertible function, is a between the elements of two , where each element of one set is paired with exactly one element of the other set, and each element of the other set is p ...

bijection
then it is called a . In the case where V = W, a linear map is called a (linear) ''
endomorphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
''. Sometimes the term linear operator 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 V and W are
real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
vector spaces (not necessarily with V = W), or it can be used to emphasize that V is a
function space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, which is a common convention in
functional analysis 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. Functional analysis is a branch of mathemat ...
. Sometimes the term ''
linear function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...

linear function
'' has the same meaning as ''linear map'', while in
analysis Analysis is the process of breaking a complexity, complex topic or Substance theory, 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 Ari ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
s in ''V'' onto linear subspaces in ''W'' (possibly of a lower
dimension In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
); for example, it maps a
plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early flying machines include all forms of aircraft studied ...
through the
origin Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * , a Wolverine comic book mini-series published by Marvel Comics in 2002 * , a 1999 ''Buffy the Vampire Slayer'' comic book series * , a major ''Judge Dred ...
in ''V'' to either a plane through the origin in ''W'', a
line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', a 2009 independent film by Nancy Schwartzman Lite ...

line
through the origin in ''W'', or just the origin in ''W''. Linear maps can often be represented as
matrices Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics) In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of numbers, symbol (formal), symbols, or expression (mathema ...
, and simple examples include rotation and reflection linear transformations. In the language of
category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
, linear maps are the
morphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

morphism
s of vector spaces.


Definition and first consequences

Let V and W 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 grassl ...
K. A function f: V \to W is said to be a ''linear map'' if for any two vectors \mathbf, \mathbf \in V and any scalar c \in K the following two conditions are satisfied: *
Additivity Interaction is a kind of action that occurs as two or more objects have an effect upon one another. The idea of a two-way effect is essential in the concept of interaction, as opposed to a one-way Causality, causal effect. Closely related terms ...
/ operation of addition f(\mathbf + \mathbf) = f(\mathbf) + f(\mathbf) *
Homogeneity Homogeneity and heterogeneity are concepts often used in the sciences and statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a sc ...
of degree 1 / operation of scalar multiplication f(c \mathbf) = c f(\mathbf) 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 \mathbf_1, \ldots, \mathbf_n \in V and scalars c_1, \ldots, c_n \in K, the following equality holds: f(c_1 \mathbf_1 + \cdots + c_n \mathbf_n) = c_1 f(\mathbf_1) + \cdots + c_n f(\mathbf_n). Denoting the zero elements of the vector spaces V and W by \mathbf_V and \mathbf_W respectively, it follows that f(\mathbf_V) = \mathbf_W. Let c = 0 and \mathbf \in V in the equation for homogeneity of degree 1: f(\mathbf_V) = f(0\mathbf) = 0f(\mathbf) = \mathbf_W. Occasionally, V and W can be vector spaces over different fields. It is then necessary to specify which of these ground fields is being used in the definition of "linear". If V and W are spaces over the same field K as above, then we talk about K-linear maps. For example, the
conjugation Conjugation or conjugate may refer to: Linguistics * Grammatical conjugation, the modification of a verb from its basic form * Emotive conjugation or Russell's conjugation, the use of loaded language Mathematics * Complex conjugation, the change ...
of
complex numbers In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
is an map \Complex \to \Complex, but it is not where \R and \Complex are symbols representing the sets of real numbers and complex numbers, respectively. A linear map V \to K with K viewed as a one-dimensional vector space over itself is called a
linear functional In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. These statements generalize to any left-module _R M over a ring R 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 f: \mathbb \to \mathbb: x \mapsto cx, of which the
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...

graph
is a line through the origin. * More generally, any
homothety
homothety
\mathbf \mapsto c\mathbf where c centered in the origin of a vector space is a linear map. * The zero map \mathbf x \mapsto \mathbf 0 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 grassl ...
) is linear. * The
identity map
identity map
on any module is a linear operator. * For real numbers, the map x \mapsto x^2 is not linear. * For real numbers, the map x \mapsto x + 1 is not linear (but is an
affine transformation In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method con ...
). * If A is a m \times n
real matrix In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
, then A defines a linear map from \R^n to \R^m by sending a
column vector 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 th ...
\mathbf x \in \R^n to the column vector A \mathbf x \in \R^m. 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 (linear algebra), basis of ''V'' over its base Field (mathematics), field. p. 44, §2.36 It is sometimes called Hamel dimension (after ...
vector spaces can be represented in this manner; see the , below. * If f: V \to W is an
isometry In mathematics, an isometry (or congruence (geometry), congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be Bijection, bijective. "We shall find it convenient to use the wor ...
between real
normed space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s such that f(0) = 0 then f is a linear map. This result is not necessarily true for complex normed space. *
Differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Product differentiation, in marketing * Differentiated service, a service that varies with the identity o ...

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 is a smooth function with compact support. In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over ...

smooth function
s (a linear operator is a linear
endomorphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
, that is a linear map where the
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 *Doma ...
and
codomain In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

codomain
of it is the same). An example is \frac \left( c_1 f_1(x) + c_2 f_2(x) + \cdots + c_n f_n(x) \right) = c_1 \frac + c_2 \frac + \cdots + c_n \frac. * A definite
integral In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

integral
over some interval is a linear map from the space of all real-valued integrable functions on to \R. For example, \int_a^b = + + \cdots + . * An indefinite
integral In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

integral
(or
antiderivative In calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zer ...
) with a fixed integration starting point defines a linear map from the space of all real-valued integrable functions on \R to the space of all real-valued, differentiable functions on \R. Without a fixed starting point, the antiderivative maps to the quotient space of the differentiable functions by the linear space of constant functions. * If V and W are finite-dimensional vector spaces over a field , of respective dimensions and , then the function that maps linear maps f: V \to W to matrices in the way described in (below) is a linear map, and even a
linear isomorphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. * The
expected value In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by exp ...
of a
random variable A random variable is a variable whose values depend on outcomes of a random In common parlance, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no ...

random variable
(which is in fact a function, and as such a element of a vector space) is linear, as for random variables X and Y we have E + Y= E + E /math> and E
X
X
= aE /math>, but the
variance In probability theory Probability theory is the branch of concerned with . Although there are several different , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of . Typically these ax ...

variance
of a random variable is not linear. File:Streckung eines Vektors.gif, The function f:\R^2 \to \R^2 with f(x, y) = (2x, y) is a linear map. This function scales the x component of a vector by the factor 2. File:Streckung der Summe zweier Vektoren.gif, The function f(x, y) = (2x, y) is additive: It doesn't matter whether vectors are first added and then mapped or whether they are mapped and finally added: f(\mathbf a + \mathbf b) = f(\mathbf a) + f(\mathbf b) File:Streckung homogenitaet Version 3.gif, The function f(x, y) = (2x, y) is homogeneous: It doesn't matter whether a vector is first scaled and then mapped or first mapped and then scaled: f(\lambda \mathbf a) = \lambda f(\mathbf a)


Matrices

If V and W are
finite-dimensional In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e. the number of vectors) of a Basis (linear algebra), basis of ''V'' over its base Field (mathematics), field. p. 44, §2.36 It is sometimes called Hamel dimension (after ...
vector spaces and a
basis Basis may refer to: Finance and accounting *Adjusted basisIn tax accounting, adjusted basis is the net cost of an asset after adjusting for various tax-related items. Adjusted Basis or Adjusted Tax Basis refers to the original cost or other b ...
is defined for each vector space, then every linear map from V to W can be represented by a
matrix Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols, or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the material in between a eukaryoti ...
. This is useful because it allows concrete calculations. Matrices yield examples of linear maps: if A is a real m \times n matrix, then f(\mathbf x) = A \mathbf x describes a linear map \R^n \to \R^m (see
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
). Let \ be a basis for V. Then every vector \bold \in V is uniquely determined by the coefficients \bold_1, \ldots , \bold_n in the field \R^n: c_1 \mathbf_1 + \cdots + c_n \mathbf_n. If f: V \to W is a linear map, f\left(c_1 \mathbf_1 + \cdots + c_n \mathbf_n\right) = c_1 f\left(\mathbf_1\right) + \cdots + c_n f\left(\mathbf_n\right), which implies that the function ''f'' is entirely determined by the vectors f(\bold_1), \ldots , f(\bold_n). Now let \ be a basis for W. Then we can represent each vector f(\bold_j) as f\left(\mathbf_j\right) = a_ \mathbf_1 + \cdots + a_ \mathbf_m. Thus, the function f is entirely determined by the values of a_. If we put these values into an m \times n matrix M, then we can conveniently use it to compute the vector output of f for any vector in V. To get M, every column j of M is a vector \begin a_ \\ \vdots \\ a_ \end corresponding to f(\bold_j) as defined above. To define it more clearly, for some column j that corresponds to the mapping f(\bold_j), \mathbf = \begin \ \cdots & a_ & \cdots\ \\ & \vdots & \\ & a_ & \end where M is the matrix of f. In other words, every column j = 1, \ldots, n has a corresponding vector f(\bold_j) whose coordinates a_ + \cdots + a_ are the elements of column j. A single linear map may be represented by many matrices. This is because the values of the elements of a matrix depend on the bases chosen. The matrices of a linear transformation can be represented visually: # Matrix for T relative to B: A # Matrix for T relative to B': A' # Transition matrix from B' to B: P # Transition matrix from B to B': P^ Such that starting in the bottom left corner \left mathbf\right and looking for the bottom right corner \left \left(\mathbf\right)\right, one would left-multiply—that is, A'\left mathbf\right = \left \left(\mathbf\right)\right. The equivalent method would be the "longer" method going clockwise from the same point such that \left mathbf\right is left-multiplied with P^AP, or P^AP\left mathbf\right = \left \left(\mathbf\right)\right.


Examples in two dimensions

In two-
dimension In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

dimension
al space R2 linear maps are described by 2 × 2
matrices Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics) In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of numbers, symbol (formal), symbols, or expression (mathema ...
. These are some examples: *
rotation A rotation is a circular movement of an object around a center (or point) of rotation. The plane (geometry), geometric plane along which the rotation occurs is called the ''rotation plane'', and the imaginary line extending from the center an ...
** by 90 degrees counterclockwise: \mathbf = \begin 0 & -1\\ 1 & 0\end ** by an angle ''θ'' counterclockwise: \mathbf = \begin \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end *
reflectionReflection or reflexion may refer to: Philosophy * Self-reflection Science * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal r ...
** through the ''x'' axis: \mathbf = \begin 1 & 0\\ 0 & -1\end ** through the ''y'' axis: \mathbf = \begin-1 & 0\\ 0 & 1\end ** through a line making an angle ''θ'' with the origin: \mathbf = \begin\cos & \sin \\ \sin & -\cos \end *
scaling Scaling may refer to: Science and technology Mathematics and physics * Scaling (geometry), a linear transformation that enlarges or diminishes objects * Scale invariance, a feature of objects or laws that do not change if scales of length, energy ...
by 2 in all directions: \mathbf = \begin 2 & 0\\ 0 & 2\end = 2\mathbf * horizontal shear mapping: \mathbf = \begin 1 & m\\ 0 & 1\end *
squeeze mapping ''r'' = 3/2 squeeze mapping In linear algebra, a squeeze mapping is a type of linear map In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...
: \mathbf = \begin k & 0\\ 0 & \frac\end * projection onto the ''y'' axis: \mathbf = \begin 0 & 0\\ 0 & 1\end.


Vector space of linear maps

The composition of linear maps is linear: if f: V \to W and g: W \to Z are linear, then so is their
composition Composition or Compositions may refer to: Arts * Composition (dance), practice and teaching of choreography * Composition (music), an original piece of music and its creation *Composition (visual arts) The term composition means "putting togethe ...
g \circ f: V \to Z. It follows from this that the
class Class or The Class may refer to: Common uses not otherwise categorized * Class (biology), a taxonomic rank * Class (knowledge representation), a collection of individuals or objects * Class (philosophy), an analytical concept used differently f ...
of all vector spaces over a given field ''K'', together with ''K''-linear maps as
morphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

morphism
s, forms a
category Category, plural categories, may refer to: Philosophy and general uses *Categorization Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...
. The
inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when add ...
of a linear map, when defined, is again a linear map. If f_1: V \to W and f_2: V \to W are linear, then so is their
pointwiseIn mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined o ...
sum f_1 + f_2, which is defined by (f_1 + f_2)(\mathbf x) = f_1(\mathbf x) + f_2(\mathbf x). If f: V \to W is linear and \alpha is an element of the ground field K, then the map \alpha f, defined by (\alpha f)(\mathbf x) = \alpha (f(\mathbf x)), is also linear. Thus the set \mathcal(V, W) of linear maps from V to W itself forms a vector space over K, sometimes denoted \operatorname(V, W). Furthermore, in the case that V = W, this vector space, denoted \operatorname(V), is an
associative algebra In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
under
composition of maps In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
, since the composition of two linear maps is again a linear map, and the composition of maps is always associative. This case is discussed in more detail below. Given again the finite-dimensional case, if bases have been chosen, then the composition of linear maps corresponds to the
matrix multiplication In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

matrix multiplication
, the addition of linear maps corresponds to the
matrix addition In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, and the multiplication of linear maps with scalars corresponds to the multiplication of matrices with scalars.


Endomorphisms and automorphisms

A linear transformation f : V \to V is an
endomorphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
of V; the set of all such endomorphisms \operatorname(V) together with addition, composition and scalar multiplication as defined above forms an
associative algebra In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
with identity element over the field K (and in particular a ring). The multiplicative identity element of this algebra is the
identity map
identity map
\operatorname: V \to V. An endomorphism of V that is also an
isomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

isomorphism
is called an
automorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

automorphism
of V. The composition of two automorphisms is again an automorphism, and the set of all automorphisms of V forms a
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
, the
automorphism group In mathematics, the automorphism group of an object ''X'' is the group (mathematics), group consisting of automorphisms of ''X''. For example, if ''X'' is a Dimension (vector space), finite-dimensional vector space, then the automorphism group of ' ...
of V which is denoted by \operatorname(V) or \operatorname(V). Since the automorphisms are precisely those
endomorphisms In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a group ...
which possess inverses under composition, \operatorname(V) is the group of
units Unit may refer to: Arts and entertainment * UNIT Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in ...
in the ring \operatorname(V). If V has finite dimension n, then \operatorname(V) is
isomorphic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

isomorphic
to the
associative algebra In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
of all n \times n matrices with entries in K. The automorphism group of V is
isomorphic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
to the
general linear group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
\operatorname(n, K) of all n \times n invertible matrices with entries in K.


Kernel, image and the rank–nullity theorem

If f: V \to W is linear, we define the
kernel Kernel may refer to: Computing * Kernel (operating system) In an operating system with a Abstraction layer, layered architecture, the kernel is the lowest level, has complete control of the hardware and is always in memory. In some systems it ...
and the
image An image (from la, imago) is an artifact that depicts visual perception Visual perception is the ability to interpret the surrounding environment (biophysical), environment through photopic vision (daytime vision), color vision, sco ...
or
range Range may refer to: Geography * Range (geographic)A range, in geography, is a chain of hill A hill is a landform A landform is a natural or artificial feature of the solid surface of the Earth or other planetary body. Landforms together ...
of f by \begin \ker(f) &= \ \\ \operatorname(f) &= \ \end \ker(f) is a subspace of V and \operatorname(f) is a subspace of W. The following
dimension In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

dimension
formula is known as the
rank–nullity theorem The rank–nullity theorem is a theorem 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 ...
: \dim(\ker( f )) + \dim(\operatorname( f )) = \dim( V ). The number \dim(\operatorname(f)) is also called the
rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking A ranking is a relationship between a set of items such that, for any two items, the first is either "rank ...
of f and written as \operatorname(f), or sometimes, \rho(f); p. 52, § 2.5.1 p. 90, § 50 the number \dim(\ker(f)) is called the nullity of f and written as \operatorname(f) or \nu(f). If V and W are finite-dimensional, bases have been chosen and f is represented by the matrix A, then the rank and nullity of f are equal to the rank and nullity of the matrix A, respectively.


Cokernel

A subtler invariant of a linear transformation f: V \to W is the , which is defined as \operatorname(f) := W/f(V) = W/\operatorname(f). This is the ''dual'' notion to the kernel: just as the kernel is a ''sub''space of the ''domain,'' the co-kernel is a ''quotient'' space of the ''target.'' Formally, one has the
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0 \to \ker(f) \to V \to W \to \operatorname(f) \to 0. These can be interpreted thus: given a linear equation ''f''(v) = w to solve, * the kernel is the space of ''solutions'' to the ''homogeneous'' equation ''f''(v) = 0, and its dimension is the number of
degrees of freedom Degrees of Freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or other physical ...
in the space of solutions, if it is not empty; * the co-kernel is the space of constraints that the solutions must satisfy, and its dimension is the maximal number of independent constraints. The dimension of the co-kernel and the dimension of the image (the rank) add up to the dimension of the target space. For finite dimensions, this means that the dimension of the quotient space ''W''/''f''(''V'') is the dimension of the target space minus the dimension of the image. As a simple example, consider the map ''f'': R2 → R2, given by ''f''(''x'', ''y'') = (0, ''y''). Then for an equation ''f''(''x'', ''y'') = (''a'', ''b'') to have a solution, we must have ''a'' = 0 (one constraint), and in that case the solution space is (''x'', ''b'') or equivalently stated, (0, ''b'') + (''x'', 0), (one degree of freedom). The kernel may be expressed as the subspace (''x'', 0) < ''V'': the value of ''x'' is the freedom in a solution – while the cokernel may be expressed via the map ''W'' → R, (a, b) \mapsto (a): given a vector (''a'', ''b''), the value of ''a'' is the ''obstruction'' to there being a solution. An example illustrating the infinite-dimensional case is afforded by the map ''f'': R → R, \left\ \mapsto \left\ with ''b''1 = 0 and ''b''''n'' + 1 = ''an'' for ''n'' > 0. Its image consists of all sequences with first element 0, and thus its cokernel consists of the classes of sequences with identical first element. Thus, whereas its kernel has dimension 0 (it maps only the zero sequence to the zero sequence), its co-kernel has dimension 1. Since the domain and the target space are the same, the rank and the dimension of the kernel add up to the same sum as the rank and the dimension of the co-kernel (\aleph_0 + 0 = \aleph_0 + 1), but in the infinite-dimensional case it cannot be inferred that the kernel and the co-kernel of an
endomorphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
have the same dimension (0 ≠ 1). The reverse situation obtains for the map ''h'': R → R, \left\ \mapsto \left\ with ''cn'' = ''a''''n'' + 1. Its image is the entire target space, and hence its co-kernel has dimension 0, but since it maps all sequences in which only the first element is non-zero to the zero sequence, its kernel has dimension 1.


Index

For a linear operator with finite-dimensional kernel and co-kernel, one may define ''index'' as: \operatorname(f) := \dim(\ker(f)) - \dim(\operatorname(f)), namely the degrees of freedom minus the number of constraints. For a transformation between finite-dimensional vector spaces, this is just the difference dim(''V'') − dim(''W''), by rank–nullity. This gives an indication of how many solutions or how many constraints one has: if mapping from a larger space to a smaller one, the map may be onto, and thus will have degrees of freedom even without constraints. Conversely, if mapping from a smaller space to a larger one, the map cannot be onto, and thus one will have constraints even without degrees of freedom. The index of an operator is precisely the
Euler characteristic#REDIRECT Euler characteristic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathema ...
of the 2-term complex 0 → ''V'' → ''W'' → 0. In
operator theoryIn mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators or ...
, the index of
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s is an object of study, with a major result being the
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.


Algebraic classifications of linear transformations

No classification of linear maps could be exhaustive. The following incomplete list enumerates some important classifications that do not require any additional structure on the vector space. Let and denote vector spaces over a field and let be a linear map.


Monomorphism

is said to be ''
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'' or a ''
monomorphism 220px In the context of abstract algebra or universal algebra, a monomorphism is an Injective function, injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of catego ...
'' if any of the following equivalent conditions are true: # is
one-to-one One-to-one or one to one may refer to: Mathematics and communication *One-to-one function, also called an injective function *One-to-one correspondence, also called a bijective function *One-to-one (communication), the act of an individual commun ...
as a map of sets. # # # is monic or left-cancellable, which is to say, for any vector space and any pair of linear maps and , the equation implies . # is left-invertible, which is to say there exists a linear map such that is the Identity function, identity map on .


Epimorphism

is said to be ''surjective'' or an ''epimorphism'' if any of the following equivalent conditions are true: # is surjective, onto as a map of sets. # # is epimorphism, epic or right-cancellable, which is to say, for any vector space and any pair of linear maps and , the equation implies . # is inverse (ring theory), right-invertible, which is to say there exists a linear map such that is the Identity function, identity map on .


Isomorphism

is said to be an ''
isomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

isomorphism
'' if it is both left- and right-invertible. This is equivalent to being both one-to-one and onto (a
bijection In , a bijection, bijective function, one-to-one correspondence, or invertible function, is a between the elements of two , where each element of one set is paired with exactly one element of the other set, and each element of the other set is p ...

bijection
of sets) or also to being both epic and monic, and so being a bimorphism. If is an endomorphism, then: * If, for some positive integer , the -th iterate of , , is identically zero, then is said to be nilpotent. * If , then is said to be idempotent * If , where is some scalar, then is said to be a scaling transformation or scalar multiplication map; see scalar matrix.


Change of basis

Given a linear map which is an
endomorphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
whose matrix is ''A'', in the basis ''B'' of the space it transforms vector coordinates [u] as [v] = ''A''[u]. As vectors change with the inverse of ''B'' (vectors are Covariance and contravariance of vectors, contravariant) its inverse transformation is [v] = ''B''[v']. Substituting this in the first expression B\left[v'\right] = AB\left[u'\right] hence \left[v'\right] = B^AB\left[u'\right] = A'\left[u'\right]. Therefore, the matrix in the new basis is ''A′'' = ''B''−1''AB'', being ''B'' the matrix of the given basis. Therefore, linear maps are said to be 1-co- 1-contra-covariance and contravariance of vectors, variant objects, or type (1, 1) tensors.


Continuity

A ''linear transformation'' between topological vector spaces, for example
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s, may be continuous function (topology), continuous. If its domain and codomain are the same, it will then be a continuous linear operator. A linear operator on a normed linear space is continuous if and only if it is bounded operator, bounded, for example, when the domain is finite-dimensional. 1.18 Theorem ''Let \Lambda be a linear functional on a topological vector space . Assume \Lambda \mathbf x \neq 0 for some \mathbf x \in X. Then each of the following four properties implies the other three:'' An infinite-dimensional domain may have discontinuous linear operators. An example of an unbounded, hence discontinuous, linear transformation is differentiation on the space of smooth functions equipped with the supremum norm (a function with small values can have a derivative with large values, while the derivative of 0 is 0). For a specific example, converges to 0, but its derivative does not, so differentiation is not continuous at 0 (and by a variation of this argument, it is not continuous anywhere).


Applications

A specific application of linear maps is for geometric transformations, such as those performed in computer graphics, where the translation, rotation and scaling of 2D or 3D objects is performed by the use of a transformation matrix. Linear mappings also are used as a mechanism for describing change: for example in calculus correspond to derivatives; or in relativity, used as a device to keep track of the local transformations of reference frames. Another application of these transformations is in compiler optimizations of nested-loop code, and in parallelizing compiler techniques.


See also

* * * * * * * *


Notes


Bibliography

* * * * * * * * * * * * * {{Authority control Abstract algebra Functions and mappings Linear operators, Transformation (function)