In mathematics, a linear map (also called a linear transformation or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. The term "linear transformation" is in particularly common use, especially for linear maps from a vector space to itself (endomorphisms). Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector. ... In mathematics, an endomorphism is a morphism (or homomorphism) from a mathematical object to itself. ...

In the language of abstract algebra, a linear map is a homomorphism of vector spaces, or a morphism in the category of vector spaces over a given field. Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures (such as groups, rings, or vector spaces). ... In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ... In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

Definition and first consequences

Let V and W be vector spaces over the same field K. A function f : V → W is said to be a linear map if for any two vectors x and y in V and any scalar a in K, the following two conditions are satisfied: In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

	additivity
	homogeneity

This is equivalent to requiring that for any vectors x₁, ..., x_m and scalars a₁, ..., a_m, the equality

holds.

Occasionally, V and W can be considered to 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 considered as spaces over the field K as above, we talk about K-linear maps. For example, the conjugation of complex numbers is an R-linear map C → C, but it is not C-linear. The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ...

A linear map from V to K (with K viewed as a vector space over itself) is called a linear functional. In linear algebra, a branch of mathematics, a linear functional or linear form is a linear function from a vector space to its field of scalars. ...

It immediately follows from the definition that f(0) = 0. Hence linear maps are sometimes called homogeneous linear maps (see linear function). A linear function is a mathematical function term of the form: f(x) = m x + c where c is a constant. ...

Examples

• The identity map and zero map are linear.

• For real numbers, the map is not linear.

• If A is an m × n matrix, then A defines a linear map from Rⁿ to R^m by sending the column vector x ∈ Rⁿ to the column vector Ax ∈ R^m. Conversely, any linear map between finite-dimensional vector spaces can be represented in this manner; see the following section.

• The integral yields a linear map from the space of all real-valued integrable functions on some interval to R

• Differentiation is a linear map from the space of all differentiable functions to the space of all functions.

• If V and W are finite-dimensional vector spaces over a field F, then functions that send linear maps f : V → W to dim_F(W)-by-dim_F(V) matrices in the way described in the sequel are themselves linear maps.

dtp hustle kidd An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ... 0 (zero), alternatively called naught, nil, nada, ought, zilch, zip, nothing or nought, is both a number and a numeral. ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... In linear algebra, a column vector is an m × 1 matrix, i. ... In mathematics, the dimension of a vector space V is the cardinality (i. ... The integral of f(x) from a to b is the area above the x-axis and below the curve y = f(x), minus the area below the x-axis and above the curve, for x in the interval [a,b]. Integration is a core concept of advanced mathematics, specifically... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... For a non-technical overview of the subject, see Calculus. ...

Matrices

If V and W are finite-dimensional, and one has chosen bases in those spaces, then every linear map from V to W can be represented as a matrix; this is useful because it allows concrete calculations. Conversely, matrices yield examples of linear maps: if A is a real m-by-n matrix, then the rule f(x) = Ax describes a linear map Rⁿ → R^m (see Euclidean space). In mathematics, the dimension of a vector space V is the cardinality (i. ... In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...

Let be a basis for V. Then every vector v in V is uniquely determined by the coefficients in

If f : V → W is a linear map,

which implies that the function f is entirely determined by the values of

Now let be a basis for W. Then we can represent the values of each f(v_j) as

Thus, the function f is entirely determined by the values of a_i,j.

If we put these values into an m-by-n matrix M, then we can conveniently use it to compute the value of f for any vector in V. For if we place the values of in an n-by-1 matrix C, we have MC = f(v).

A single linear map may be represented by many matrices. This is because the values of the elements of the matrix depend on the bases that are chosen.

Examples of linear transformation matrices

Some special cases of linear transformations of two-dimensional space R² are illuminating: 2-dimensional renderings (ie. ...

• rotation by 90 degrees counterclockwise:

  

• reflection against the x axis:

  

• scaling by 2 in all directions:

  

• vertical shear:

  

• squeezing:

  

• projection onto the y axis:

  

In the three-dimensional space, the possible moves of a rigid body are rotations and translations. ... In erik, a reflection (also spelled reflexion) is a map that transforms an object into its mirror image. ... In Euclidean geometry, uniform scaling is a linear transformation that enlarges or diminishes objects; the scale factor is the same in all directions; it is also called a homothety. ... In this shear transformation of an image of the Mona Lisa, the picture was deformed in such a way that its central vertical axis was not modified. ... In mathematics, a squeeze mapping in linear algebra is a type of linear transformation that preserves Euclidean area of regions in the cartesian plane, but is not a Euclidean motion. ... The transformation P is the orthogonal projection onto the line m. ...

Forming new linear maps from given ones

The composition of linear maps is linear: if f : V → W and g : W → Z are linear, then so is g o f : V → Z.

The inverse of a linear map, when defined, is again a linear map. In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ...

If f₁ : V → W and f₂ : V → W are linear, then so is their sum f₁ + f₂ (which is defined by (f₁ + f₂)(x) = f₁(x) + f₂(x)).

If f : V → W is linear and a is an element of the ground field K, then the map af, defined by (af)(x) = a (f(x)), is also linear.

Thus the set L(V,W) of linear maps from V to W itself forms a vector space over K, sometimes denoted Hom(V,W). Furthermore, in the case that V=W, this vector space (denoted End(V)) is an associative algebra 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. In mathematics, an associative algebra is a vector space (or more generally, a module) which also allows the multiplication of vectors in a distributive and associative manner. ... In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ...

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. This article gives an overview of the various ways to perform matrix multiplication. ... The operations on matrices differ from similar operations of scalar algebra in several respects. ...

Endomorphisms and automorphisms

A linear transformation f : V → V is an endomorphism of V; the set of all such endomorphisms End(V) together with addition, composition and scalar multiplication as defined above forms an associative algebra with identity element over the field K (and in particular a ring). The multiplicative identity element of this algebra is the identity map id : V → V. In mathematics, an endomorphism is a morphism (or homomorphism) from a mathematical object to itself. ... In mathematics, an associative algebra is a vector space (or more generally, a module) which also allows the multiplication of vectors in a distributive and associative manner. ... In ring theory, a branch of abstract algebra, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ...

An endomorphism of V that is also an isomorphism is called an automorphism 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 of V which is denoted by Aut(V) or GL(V). Since the automorphisms are precisely those endomorphisms which possess inverses under composition, Aut(V) is the group of units in the ring End(V). In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In mathematics, a unit in a ring R is an element u such that there is v in R with uv = vu = 1R. That is, u is an invertible element of the multiplicative monoid of R. The units of R form a group U(R) under multiplication, the group of...

If V has finite dimension n, then End(V) is isomorphic to the associative algebra of all n by n matrices with entries in K. The automorphism group of V is isomorphic to the general linear group GL(n, K) of all n by n invertible matrices with entries in K. In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... In mathematics, an associative algebra is a vector space (or more generally, a module) which also allows the multiplication of vectors in a distributive and associative manner. ... In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. ... In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. ...

Kernel, image and the rank-nullity theorem

If f : V → W is linear, we define the kernel and the image or range of f by In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. ... In mathematics, the image of an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with x. ... In mathematics, the range of a function is the set of all output values produced by that function. ...

ker(f) is a subspace of V and im(f) is a subspace of W. The following dimension formula, known as the rank-nullity theorem, is often useful: The concept of a linear subspace (or vector subspace) is important in linear algebra and related fields of mathematics. ... 2-dimensional renderings (ie. ... In mathematics, the rank-nullity theorem of linear algebra, in its simplest form, relates the rank and the nullity of a matrix together with the number of columns of the matrix. ...

The number dim(im(f)) is also called the rank of f and written as rk(f), or sometimes, ρ(f); the number dim(ker(f)) is called the nullity of f and written as ν(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. In linear algebra, the column rank (row rank respectively) of a matrix A with entries in some field is defined to be the maximal number of columns (rows respectively) of A which are linearly independent. ... In mathematics, the null space (also nullspace) of an operator A is the set of all operands v which solve the equation Av = 0. ...

Algebraic classifications of linear transformations

No classification of linear maps could hope to be exhaustive. The following incomplete list enumerates some important classifications that do not require any additional structure on the vector space.

Let V and W denote vector spaces over a field, F. Let T:V → W be a linear map.

• T is said to be injective or a monomorphism if any of the following equivalent conditions are true:
  • T is one-to-one as a map of sets.
  • ker T = 0
  • T is monic or left-cancellable, which is to say, for any vector space U and any pair of linear maps R:U → V and S:U → V, the equation TR=TS implies R=S.
  • T is left-invertible, which is to say there exists a linear map S:W → V such that ST is the identity map on V.

• T is said to be surjective or an epimorphism if any of the following equivalent conditions are true:
  • T is onto as a map of sets.
  • coker T = 0
  • T is epic or right-cancellable, which is to say, for any vector space U and any pair of linear maps R:W → U and S:W → U, the equation RT=ST implies R=S.
  • T is right-invertible, which is to say there exists a linear map S:W → V such that TS is the identity map on V.

• T is said to be an isomorphism if it is both left- and right-invertible. This is equivalent to T being both one-to-one and onto (a bijection of sets) or also to T being both epic and monic, and so being a bimorphism.

• If T: V → V is an endomorphism, then:
  • If, for some positive integer n, the n-th iterate of T, Tⁿ, is identically zero, then T is said to be nilpotent.
  • If T T = T, then T is said to be idempotent
  • If T = k I, where k is some scalar, then T is said to be a scaling transformation or scalar multiplication map.

In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... In the context of abstract algebra or universal algebra, a monomorphism is simply an injective homomorphism. ... In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... The word set, which is among the words with the most numerous definitions in the English language (at 464 definitions according to the Oxford English Dictionary), may have one of the following meanings. ... In mathematics, a monic can refer to monic morphism – a special kind of morphism in category theory, monic polynomial – a polynomial whose leading coefficient is one. ... In mathematics, the idea of inverse element generalises the concepts of negation, in relation to addition, and reciprocal, in relation to multiplication. ... An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ... In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ... In category theory an epimorphism (also called an epic morphism or an epi) is a morphism f : X → Y which is right-cancellable in the following sense: g1 o f = g2 o f implies g1 = g2 for all morphisms g1, g2 : Y → Z. Epimorphisms are analogues of surjective functions, but... In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ... The word set, which is among the words with the most numerous definitions in the English language (at 464 definitions according to the Oxford English Dictionary), may have one of the following meanings. ... In abstract algebra, the cokernel of a homomorphism f : X → Y is the quotient of Y by the image of f. ... In category theory an epimorphism (also called an epic morphism or an epi) is a morphism f : X → Y which is right-cancellable in the following sense: g1 o f = g2 o f implies g1 = g2 for all morphisms g1, g2 : Y → Z. Epimorphisms are analogues of surjective functions, but... In mathematics, the idea of inverse element generalises the concepts of negation, in relation to addition, and reciprocal, in relation to multiplication. ... An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... A bijective function. ... In mathematics, a morphism is an abstraction of a function or mapping between two spaces. ... In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0. ... In mathematics, an idempotent element is an element which, intuitively, leaves something unchanged. ...

Continuity

Main article: Discontinuous linear map

A linear operator between topological vector spaces, for example normed spaces, may also be continuous and therefore be a continuous linear operator. On a normed space, a linear operator is continuous if and only if it is bounded, for example, when the domain is finite-dimensional. If the domain is infinite-dimensional, then there may be discontinuous linear operators. An example of an unbounded, hence not continuous, 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). In mathematics, linear maps form an important class of simple functions which preserve the algebraic structure of linear spaces and are often used as approximations to more general functions (see linear approximation). ... In mathematics a topological vector space is one of the basic structures investigated in functional analysis. ... In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can be easily extended to any real vector space Rn. ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ... In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. ... In mathematics, the operator norm is a norm defined on the space of bounded operators between two Banach spaces. ... In mathematics, linear maps form an important class of simple functions which preserve the algebraic structure of linear spaces and are often used as approximations to more general functions (see linear approximation). ...

Applications

A specific application of linear maps is in the field of computational neuroscience. An example of a system being modeled is the innervation of V1 (primary visual cortex) by the retina. This transformation is called the logmap transformation. This kind of transformation is known as a domain coordinate transformation and provides a mathematical model of how neural states can be conferred within the system (CNS and PNS), when a change of state is required, such as from the retina to V1 as previously mentioned.

Another specific application 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. For the journal by ACM SIGGRAPH, see Computer Graphics (Publication). ... In linear algebra, linear transformations can be represented by matrices. ...

Another application of these transformations is in compiler optimizations of nested loop code, and in parallelizing compiler techniques. Compiler optimization techniques are optimization techniques that have been programmed into a compiler. ...

See also

• Linear equation
• Antilinear map
• Transformation matrix
• Continuous linear operator
• wikibooks:Algebra/Linear transformations
• Neural network
• Computer graphics

A linear equation is an equation in which each term is either a constant or the product of a constant times the first power of a variable. ... In mathematics, a mapping f : V → W from a complex vector space to another is said to be antilinear (or conjugate-linear or semilinear) if for all a, b in C and all x, y in V. The composition of two antilinear maps is complex-linear. ... In linear algebra, linear transformations can be represented by matrices. ... In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. ... // See also Artificial neural network. ... For the journal by ACM SIGGRAPH, see Computer Graphics (Publication). ...

References

• Halmos, Paul R., Finite-Dimensional Vector Spaces, Springer-Verlag, (1993). ISBN 0-387-90093-4.