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Linear transformation
 
 
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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a linear map (also called a linear transformation) is a function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 between two vector space
Vector space

File:Vector addition ans scaling.pngA vector space is a mathematical structure formed by a collection of vectors: objects that may be Vector addition together and Scalar multiplication by numbers, called scalar s in this context....
s that preserves the operations of vector addition and scalar
Scalar (mathematics)

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....
 multiplication. The expression "linear operator" is in especially common use, for linear maps from a vector space to itself (endomorphisms). In advanced mathematics, the definition of linear function
Linear function

In mathematics, the term linear function can refer to either of two different but related concepts: a first-degree polynomial function of one variable; or a map between two vector spaces that preserves vector addition and scalar multiplication....
 coincides with the definition of linear map.

In the language of abstract algebra
Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as group , ring , field , module , vector spaces, and algebra over a field....
, a linear map is a homomorphism
Homomorphism

In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ???? meaning "same" and ???f? meaning "shape"....
 of vector spaces, or a morphism
Morphism

In mathematics, a morphism is an Abstraction derived from structure-preserving map between two mathematical structures.The study of morphisms and of the structures over which they are defined, is central to category theory....
 in the category
Category theory

In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from set s and function s to objects linked in diagrams by morphisms or arrows....
 of vector spaces over a given field
Field (mathematics)

In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication and division , satisfying certain axioms....
.

Definition and first consequences
Let V and W be vector spaces over the same field
Field (mathematics)

In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication and division , satisfying certain axioms....
 K.






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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a linear map (also called a linear transformation) is a function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 between two vector space
Vector space

File:Vector addition ans scaling.pngA vector space is a mathematical structure formed by a collection of vectors: objects that may be Vector addition together and Scalar multiplication by numbers, called scalar s in this context....
s that preserves the operations of vector addition and scalar
Scalar (mathematics)

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....
 multiplication. The expression "linear operator" is in especially common use, for linear maps from a vector space to itself (endomorphisms). In advanced mathematics, the definition of linear function
Linear function

In mathematics, the term linear function can refer to either of two different but related concepts: a first-degree polynomial function of one variable; or a map between two vector spaces that preserves vector addition and scalar multiplication....
 coincides with the definition of linear map.

In the language of abstract algebra
Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as group , ring , field , module , vector spaces, and algebra over a field....
, a linear map is a homomorphism
Homomorphism

In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ???? meaning "same" and ???f? meaning "shape"....
 of vector spaces, or a morphism
Morphism

In mathematics, a morphism is an Abstraction derived from structure-preserving map between two mathematical structures.The study of morphisms and of the structures over which they are defined, is central to category theory....
 in the category
Category theory

In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from set s and function s to objects linked in diagrams by morphisms or arrows....
 of vector spaces over a given field
Field (mathematics)

In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication and division , satisfying certain axioms....
.

Definition and first consequences


Let V and W be vector spaces over the same field
Field (mathematics)

In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication and division , satisfying certain axioms....
 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:

 additivity
Additive function

Different definitions exist depending on the specific field of application. Traditionally, an additive function is a function that preserves the addition operation:for any two elements x and y in the domain....
 homogeneity of degree 1
Homogeneous function

In mathematics, a homogeneous function is a function with multiplicative scaling behaviour: if the argument is multiplied by a factor, then the result is multiplied by some power of this factor....


This is equivalent to requiring that for any vectors x1, ..., xm and scalars a1, ..., am, the equality holds.

It immediately follows from the definition that f(0) = 0.

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
Complex conjugate

In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. Thus, the conjugate of the complex number...
 of complex numbers is an R-linear map C ? C, but it is not C-linear.

A linear map from V to K (with K viewed as a vector space over itself) is called a linear functional
Linear functional

In linear algebra, a branch of mathematics, a linear functional or linear form is a linear map from a vector space to its field of scalar s....
.

Examples


  • The identity map
    Identity function

    In mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that was used as its argument....
     and zero map are linear.
  • For real numbers, the map is not linear.
  • For real numbers, the map is not linear.
  • If A is an m × n matrix
    Matrix (mathematics)

    In mathematics, a matrix is a rectangular array of numbers, as shown at the right. In addition to a number of elementary, entrywise operations such as matrix addition a key notion is matrix multiplication....
    , then A defines a linear map from Rn to Rm by sending the column vector
    Column vector

    In linear algebra, a column vector or column matrix is an m × 1 matrix , i.e. a matrix consisting of a single column of elements....
     x ? Rn to the column vector Ax ? Rm. Conversely, any linear map between finite-dimensional vector spaces can be represented in this manner; see the following section.
  • The integral
    Integral

    Integration is an important concept in mathematics, specifically in the field of calculus and, more broadly, mathematical analysis. Given a function ƒ of a Real number variable x and an interval [a, b] of the real line, the integral...
     yields a linear map from the space of all real-valued integrable functions on some interval
    Interval (mathematics)

    In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set....
     to R
  • Differentiation
    Derivative

    In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point....
     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 dimF(W)-by-dimF(V) matrices in the way described in the sequel are themselves linear maps.


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
Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, as shown at the right. In addition to a number of elementary, entrywise operations such as matrix addition a key notion is matrix multiplication....
; 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 ? Rm (see Euclidean space
Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then 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 as Thus, the function f is entirely determined by the values of

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 = the m-by-1 matrix whose i.th element is the coordinate of f(v) which belongs to the base .

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-dimension
Dimension

In mathematics, the dimension of a space is roughly defined as the minimum number of coordinates needed to specify every point within it. For example: a point on the unit circle in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate , so the circle is 1-dimensional even though it exists in...
al space R2 are illuminating:
  • rotation
    Rotation (mathematics)

    In geometry and linear algebra, a rotation is a transformation in a plane or in space that describes the motion of a rigid body around a fixed point....
     by 90 degrees clockwise:* rotation
    Rotation (mathematics)

    In geometry and linear algebra, a rotation is a transformation in a plane or in space that describes the motion of a rigid body around a fixed point....
     by ? degrees clockwise:*reflection
    Reflection (mathematics)

    In mathematics, a reflection is a function that transforms an object into its mirror image. For example, a reflection of the small English letter p in respect to a vertical line would look like q....
     against the x axis:*scaling
    Scaling (geometry)

    In Euclidean geometry, uniform scaling or isotropic scaling is a linear transformation that enlarges or increases or diminishes objects; the scale factor is the same in all directions; it is also called a homothety....
     by 2 in all directions:*vertical shear mapping:* squeezing
    Squeeze mapping

    In linear algebra, a squeeze mapping is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is not a Euclidean motion....
    :* projection
    Projection (linear algebra)

    In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that P2 = P....
     onto the y axis:

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
Inverse function

In mathematics, if ƒ is a function from A to B then an inverse function for ƒ is a function in the opposite direction, from B to A, with the property that a round trip from A to B to A returns each element of the initial set to itself....
 of a linear map, when defined, is again a linear map.

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
Associative algebra

In mathematics, an associative algebra is a vector space which also allows the multiplication of vectors in a distributivity and associativity manner....
 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
Matrix multiplication

In mathematics, matrix multiplication is the operation of multiplying a matrix with either a scalar or another matrix. This article gives an overview of the various ways to perform matrix multiplication....
, the addition of linear maps corresponds to the matrix addition
Matrix addition

In mathematics, matrix addition is the operation of adding two matrix by adding the corresponding entries together. However, there is another operation which could also be considered as a kind of addition for matrices....
, and the multiplication of linear maps with scalars corresponds to the multiplication of matrices with scalars.

Endomorphisms and automorphisms

A linear transformation f : V ? V is an endomorphism
Endomorphism

In mathematics, an endomorphism is a morphism from a mathematical object to itself. For example, an endomorphism of a vector space V is a linear map ?: V ? V, and an endomorphism of a group G is a group homomorphism ?: G ? G....
 of V; the set of all such endomorphisms End(V) together with addition, composition and scalar multiplication as defined above forms an associative algebra
Associative algebra

In mathematics, an associative algebra is a vector space which also allows the multiplication of vectors in a distributivity and associativity manner....
 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

An identity map is a database access design pattern used to improve performance by providing a context-specific in-memory cache to prevent duplicate retrieval of the same object data from the database....
 id : V ? V.

An endomorphism of V that is also an isomorphism
Isomorphism

In abstract algebra, an isomorphism is a bijection map f such that both f and its inverse function f −1 are homomorphisms, i.e., structure-preserving mappings....
 is called an automorphism
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 map the object to itself while preserving all of its structure....
 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
Unit (ring theory)

In mathematics, a unit in a ring R is an invertible element of R, i.e. an element u such that there is a v in R withThat is, u is an invertible element of the multiplicative monoid of R....
 in the ring End(V).

If V has finite dimension n, then End(V) is isomorphic
Isomorphism

In abstract algebra, an isomorphism is a bijection map f such that both f and its inverse function f −1 are homomorphisms, i.e., structure-preserving mappings....
 to the associative algebra
Associative algebra

In mathematics, an associative algebra is a vector space which also allows the multiplication of vectors in a distributivity and associativity manner....
 of all n by n matrices with entries in K. The automorphism group of V is isomorphic
Group isomorphism

In abstract algebra, a group isomorphism is a function between two group s that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations....
 to the general linear group
General linear group

In mathematics, the general linear group of degree n is the set of n×n invertible matrix, together with the operation of ordinary matrix multiplication....
 GL(n, K) of all n by n invertible matrices with entries in K.

Kernel, image and the rank-nullity theorem

If f : V ? W is linear, we define the kernel
Kernel (linear operator)

In linear algebra and functional analysis, the kernel of a linear operator L is the set of all operands v for which L = 0....
 and the image
Image (mathematics)

In mathematics, the image of a set under a given function is the set of all possible function outputs when taking each element of the set, successively, as the function's argument....
 or range
Range (mathematics)

In mathematics, the range of a function is the Set of all "output" values produced by that function. Sometimes it is called the , or more precisely, the image of the domain of the function....
 of f by ker(f) is a subspace
Linear subspace

The concept of a linear subspace is important in linear algebra and related fields of mathematics.A linear subspace is usually called simply a subspace when the context serves to distinguish it from other kinds of subspaces....
 of V and im(f) is a subspace of W. The following dimension
Dimension

In mathematics, the dimension of a space is roughly defined as the minimum number of coordinates needed to specify every point within it. For example: a point on the unit circle in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate , so the circle is 1-dimensional even though it exists in...
 formula, known as the rank-nullity theorem
Rank-nullity theorem

In mathematics, the rank?nullity theorem of linear algebra, in its simplest form, states that the rank and the nullity of a matrix add up to the number of columns of the matrix....
, is often useful:

The number dim(im(f)) is also called the rank of f and written as rk(f), or sometimes, ?(f); the number dim(ker(f)) is called the nullity of f and written as ?(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
Null space

In linear algebra, the kernel or null space of a matrix A is the set of all vectors x for which Ax = 0....
 of the matrix A, respectively.

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
    Monomorphism

    In the context of abstract algebra or universal algebra, a monomorphism is an Injective function homomorphism. A monomorphism from X to Y is often denoted with the notation ....
     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
      Monic

      In mathematics, monic can refer to*monic morphism - a special kind of morphism in category theory.*monic polynomial - a polynomial whose leading coefficient is one....
       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
      Identity function

      In mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that was used as its argument....
       on V.


  • T is said to be surjective or an epimorphism
    Epimorphism

    In category theory an epimorphism is a morphism f : X ? Y which is Cancellation property in the sense that, for all morphisms ,Epimorphisms are analogues of surjective functions, but they are not exactly the same....
     if any of the following equivalent conditions are true:
    • T is onto as a map of sets.
    • coker
      Cokernel

      In mathematics, the cokernel of a linear mapping of vector spaces f : X ? Y is the quotient space Y/im of the codomain of f by the image of f....
       T = 0
    • T is epic
      Epimorphism

      In category theory an epimorphism is a morphism f : X ? Y which is Cancellation property in the sense that, for all morphisms ,Epimorphisms are analogues of surjective functions, but they are not exactly the same....
       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
      Identity function

      In mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that was used as its argument....
       on V.


  • T is said to be an isomorphism
    Isomorphism

    In abstract algebra, an isomorphism is a bijection map f such that both f and its inverse function f −1 are homomorphisms, i.e., structure-preserving mappings....
     if it is both left- and right-invertible. This is equivalent to T being both one-to-one and onto (a bijection
    Bijection

    In mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property that, for every y in Y, there is exactly one x in X such that f = y....
     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
      Nilpotent

      In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0....
      .
    • 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.


Continuity


A linear operator between topological vector space
Topological vector space

In mathematics, a topological vector space is one of the basic structures investigated in functional analysis. As the name suggests the space blends a topology with the algebraic concept of a vector space....
s, for example normed spaces, may also be continuous
Continuous function (topology)

In topology and related areas of mathematics a continuous function is a morphism between topological spaces. Intuitively, this is a function f where a set of points near f always contain the of a set of points near x....
 and therefore be a continuous linear operator
Continuous linear operator

In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous function linear transformation between topological vector spaces....
. On a normed space, a linear operator is continuous if and only if it is bounded
Bounded operator

In functional analysis , a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L to that of v is bounded set by the same number, over all non-zero vectors v in X....
, 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, while the derivative of 0 is 0).

Applications


A specific application of linear maps is for geometric transformations, such as those performed in computer graphics
Computer graphics

Computer graphics are graphics created by computers and, more generally, the representation and manipulation of pictorial data by a computer....
, where the translation, rotation and scaling of 2D or 3D objects is performed by the use of a transformation matrix
Transformation matrix

In linear algebra, linear transformations can be represented by matrix . If T is a linear transformation mapping Rn to Rm and x is a column vector with n entries, then...
.

Another application of these transformations is in compiler optimizations of nested-loop code, and in parallelizing compiler techniques.

See also

  • Affine transformation
    Affine transformation

    In geometry, an affine transformation or affine map or an affinity between two vector spaces consists of a linear transformation followed by a translation :...
  • Linear equation
    Linear equation

    A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable.Linear equations can have one or more variables....
  • Antilinear map
    Antilinear map

    In mathematics, a map f : V ? W from a complex vector space to another is said to be antilinear iffor all a, b in C and all x, y in V....
  • Transformation matrix
    Transformation matrix

    In linear algebra, linear transformations can be represented by matrix . If T is a linear transformation mapping Rn to Rm and x is a column vector with n entries, then...
  • Continuous linear operator
    Continuous linear operator

    In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous function linear transformation between topological vector spaces....
  • wikibooks:Algebra/Linear transformations
  • Neural network
    Neural network

    Traditionally, the term neural network had been used to refer to a network or circuit of neuron. The modern usage of the term often refers to artificial neural networks, which are composed of artificial neurons or nodes....
  • Computer graphics
    Computer graphics

    Computer graphics are graphics created by computers and, more generally, the representation and manipulation of pictorial data by a computer....