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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 V \to W 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 V = W, 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 V and W are real vector spaces (not necessarily with V = W), or it can be used to emphasize that V 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 V and W be vector spaces over the same field 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 / operation of addition f(\mathbf + \mathbf) = f(\mathbf) + f(\mathbf) *
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 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). Thus a linear map is one which preserves linear combinations. 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. A linear map V \to K with K viewed as a one-dimensional vector space over itself is called a linear functional. 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 is a line through the origin. * More generally, any 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) 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 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, 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 A is a m \times n real matrix, then A defines a linear map from \R^n to \R^m by sending a column vector \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 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 f: V \to W 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 f(0) = 0 then f 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 \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, 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 \R. For example, \int_a^b \left _1 f_1(x) + c_2 f_2(x) + \dots + c_n f_n(x)\right\, dx = + c_2 \int_a^b f_2(x) \, dx + \cdots + c_n \int_a^b f_n(x) \, dx. * 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 \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, 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 X and Y we have E + Y= E + E /math> and E X= aE /math>, but the
variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
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)


Linear extensions

Often, a linear map is constructed by defining it on a subset of a vector space and then to the
linear span In mathematics, the linear span (also called the linear hull or just span) of a set of vectors (from a vector space), denoted , pp. 29-30, §§ 2.5, 2.8 is defined as the set of all linear combinations of the vectors in . It can be characterized ...
of the domain. A ' of a function f is an
extension Extension, extend or extended may refer to: Mathematics Logic or set theory * Axiom of extensionality * Extensible cardinal * Extension (model theory) * Extension (predicate logic), the set of tuples of values that satisfy the predicate * Ext ...
of f to some
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 ...
that is a linear map. Suppose X and Y are vector spaces and f : S \to Y is a function defined on some subset S \subseteq X. Then f can be extended to a linear map F : \operatorname S \to Y if and only if whenever n > 0 is an integer, c_1, \ldots, c_n are scalars, and s_1, \ldots, s_n \in S are vectors such that 0 = c_1 s_1 + \cdots + c_n s_n, then necessarily 0 = c_1 f\left(s_1\right) + \cdots + c_n f\left(s_n\right). If a linear extension of f : S \to Y exists then the linear extension F : \operatorname S \to Y is unique and F\left(c_1 s_1 + \cdots c_n s_n\right) = c_1 f\left(s_1\right) + \cdots + c_n f\left(s_n\right) holds for all n, c_1, \ldots, c_n, and s_1, \ldots, s_n as above. If S is linearly independent then every function f : S \to Y into any vector space has a linear extension to a (linear) map \;\operatorname S \to Y (the converse is also true). For example, if X = \R^2 and Y = \R then the assignment (1, 0) \to -1 and (0, 1) \to 2 can be linearly extended from the linearly independent set of vectors S := \ to a linear map on \operatorname\ = \R^2. The unique linear extension F : \R^2 \to \R is the map that sends (x, y) = x (1, 0) + y (0, 1) \in \R^2 to F(x, y) = x (-1) + y (2) = - x + 2 y. Every (scalar-valued) linear functional f defined on a vector subspace of a real or complex vector space X has a linear extension to all of X. Indeed, the Hahn–Banach dominated extension theorem even guarantees that when this linear functional f is dominated by some given
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk a ...
p : X \to \R (meaning that , f(m), \leq p(m) holds for all m in the domain of f) then there exists a linear extension to X that is also dominated by p.


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 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 and a basis is defined for each vector space, then every linear map from V to W can be represented by a
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
. 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 geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
). Let \ be a basis for V. Then every vector \mathbf \in V is uniquely determined by the coefficients c_1, \ldots , c_n in the field \R: \mathbf = c_1 \mathbf_1 + \cdots + c_n \mathbf _n. If f: V \to W is a linear map, f(\mathbf) = f(c_1 \mathbf_1 + \cdots + c_n \mathbf_n) = c_1 f(\mathbf_1) + \cdots + c_n f\left(\mathbf_n\right), which implies that the function ''f'' is entirely determined by the vectors f(\mathbf _1), \ldots , f(\mathbf _n). Now let \ be a basis for W. Then we can represent each vector f(\mathbf _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(\mathbf _j) as defined above. To define it more clearly, for some column j that corresponds to the mapping f(\mathbf _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(\mathbf _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 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 ...
al space R2 linear maps are described by 2 × 2 matrices. These are some examples: *
rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
** 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 * reflection ** 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\cos2\theta & \sin2\theta \\ \sin2\theta & -\cos2\theta \end * scaling 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: \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 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 ...
of all vector spaces over a given field ''K'', together with ''K''-linear maps as
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, forms a category. 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 a ...
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
pointwise In 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 ...
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, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
under composition of maps, 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, the addition of linear maps corresponds to the matrix addition, 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 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, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
with identity element over the field K (and in particular a ring). The multiplicative identity element of this algebra is 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 ...
\operatorname: V \to V. An endomorphism of V that is also an
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
is called an
automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphis ...
of V. The composition of two automorphisms is again an automorphism, and the set of all automorphisms of V forms a group, the
automorphism group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
of V which is denoted by \operatorname(V) or \operatorname(V). Since the automorphisms are precisely those endomorphisms which possess inverses under composition, \operatorname(V) is the group of units in the ring \operatorname(V). If V has finite dimension n, then \operatorname(V) is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to the
associative algebra In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
of all n \times n matrices with entries in K. The automorphism group of V is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to the general linear group \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 and the
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...
or range 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 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 ...
formula is known as the rank–nullity theorem: \dim(\ker( f )) + \dim(\operatorname( f )) = \dim( V ). The number \dim(\operatorname(f)) is also called the 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 ''co''kernel, 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 exact sequence 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 thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
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 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 In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological spac ...
of the 2-term complex 0 → ''V'' → ''W'' → 0. In operator theory, the index of Fredholm operators is an object of study, with a major result being the
Atiyah–Singer index theorem In differential geometry, the Atiyah–Singer index theorem, proved by Michael Atiyah and Isadore Singer (1963), states that for an elliptic differential operator on a compact manifold, the analytical index (related to the dimension of the sp ...
.


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 ''
injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...
'' or a '' monomorphism'' if any of the following equivalent conditions are true: # is one-to-one 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 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 .


Epimorphism

is said to be ''
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element o ...
'' or an '' epimorphism'' if any of the following equivalent conditions are true: # is onto as a map of sets. # # is epic or right-cancellable, which is to say, for any vector space and any pair of linear maps and , the equation implies . # is right-invertible, which is to say there exists a linear map such that is 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 .


Isomorphism

is said to be an ''
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
'' if it is both left- and right-invertible. This is equivalent to being both one-to-one and onto (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 ...
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 In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the cl ...
. * 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 whose matrix is ''A'', in the basis ''B'' of the space it transforms vector coordinates as = ''A'' As vectors change with the inverse of ''B'' (vectors are contravariant) its inverse transformation is = ''B'' ' Substituting this in the first expression B\left '\right= AB\left '\right/math> hence \left '\right= B^AB\left '\right= A'\left '\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- variant objects, or type (1, 1)
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
s.


Continuity

A ''linear transformation'' between topological vector spaces, for example
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, may be
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
. 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, 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 transformation In mathematics, a geometric transformation is any bijection of a set to itself (or to another such set) with some salient geometrical underpinning. More specifically, it is a function whose domain and range are sets of points — most often b ...
s, such as those performed in
computer graphics Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...
, 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 Transformation (function)