Linear transformation

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Linear transformation

In mathematicsMathematics

Mathematics is the discipline that deals with concepts such as quantity, structure, space and change....
, a linear map (also called a linear transformation, or linear operator) is a functionFunction (mathematics)

In mathematics, a function relates each of its inputs to exactly one output....
between two vector spaceVector space

In mathematics, a vector space is a collection of objects that, informally speaking, may be scaled and added....
s that preserves the operations of vector addition and scalarScalar (mathematics)

In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar m...
multiplication. The expression "linear transformation" is in particularly common use, especially for linear maps from a vector space to itself. In advanced mathematics, the definition of linear functionLinear function

A linear function can refer to two slightly different concepts....
coincides with the definition of linear map.

In the language of abstract algebraAbstract algebra

Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vect...
, a linear map is a homomorphismHomomorphism

In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures ....
of vector spaces, or a morphismMorphism

In mathematics, a morphism is an abstraction of a structure-preserving mapping between two mathematical structures....
in the categoryCategory theory

In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them....
of vector spaces over a given fieldField (mathematics)

In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and ...
.

Definition and first consequences

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

In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and ...
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:

	additivityAdditive function Different definitions exist depending on the specific field of application....
	homogeneity of degree 1Homogeneous function In mathematics, a homogeneous function is a function with multiplicative scaling behaviour: if the argument is multiplied by...

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

In mathematics, the complex conjugate...
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 functionalLinear functional

In linear algebra, a branch of mathematics, a linear functional or linear form is a linear function from a vector spac...
.

Examples

The identity mapIdentity function

In mathematics, an identity function, also called identity map or identity transformation, is a function which d...
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 matrixMatrix (mathematics)

In mathematics, a matrix is a rectangular table of numbers or, more generally, a table consisting of abstract quantities tha...
, then A defines a linear map from Rⁿ to R^m by sending the column vectorColumn vector

In linear algebra, a column vector is an m × 1 matrix, i.e....
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 integralIntegral Summary

In calculus, the integral of a function is an extension of the concept of a sum....
yields a linear map from the space of all real-valued integrable functions on some intervalInterval (mathematics)

In elementary algebra, an interval is a set that contains every real number between two indicated numbers and possibly the t...
to R

DifferentiationDerivative Summary

In mathematics, the derivative is defined as the instantaneous rate of change of a function....
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.

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 matrixMatrix (mathematics)

In mathematics, a matrix is a rectangular table of numbers or, more generally, a table consisting of abstract quantities tha...
; 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 spaceEuclidean space Summary

Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called "plane Euclidean geometry", wh...
).

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 = 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-dimensionDimension

In common usage, a dimension is a parameter or measurement required to define the characteristics of an object—i.e....
al space R² are illuminating:

rotationRotation (mathematics)

In linear algebra and geometry, a rotation is a type of transformation from one system of coordinates to another system of c...
by 90 degrees counterclockwise:* rotationRotation (mathematics)

In linear algebra and geometry, a rotation is a type of transformation from one system of coordinates to another system of c...
by ? degrees counterclockwise:*reflectionReflection (mathematics)

In mathematics, a reflection is a map that transforms an object into its mirror image....
against the x axis:*scalingScaling (geometry)

In Euclidean geometry, uniform scaling is a linear transformation that enlarges or diminishes objects; the scale factor is t...
by 2 in all directions:*vertical shear mapping:* squeezingSqueeze mapping Overview

In mathematics, a squeeze mapping in linear algebra is a type of linear transformation that preserves Euclidean area of regi...
:* projectionFacts About Projection (linear algebra)

In linear algebra, a projection is a linear transformation P such that P2 = P, in other words an idempotent tran...
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 inverseInverse function

In mathematics, an inverse function is in simple terms a function which "does the reverse" of a given function....
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 algebraAssociative algebra

In mathematics, an associative algebra is a vector space which also allows the multiplication of vectors in a distributive ...
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 multiplicationMatrix multiplication

This article gives an overview of the various ways to multiply matrices. ...
, the addition of linear maps corresponds to the matrix additionMatrix addition

In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together....
, 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 endomorphismEndomorphism

In mathematics, an endomorphism is a morphism from a mathematical object to itself....
of V; the set of all such endomorphisms End(V) together with addition, composition and scalar multiplication as defined above forms an associative algebraFacts About Associative algebra

In mathematics, an associative algebra is a vector space which also allows the multiplication of vectors in a distributive ...
with identity element over the field K (and in particular a ring). The multiplicative identity element of this algebra is the identity mapIdentity map Summary

An identity map is a database access design pattern used to improve performance by providing a context-specific in-memory ca...
id : V ? V.

An endomorphism of V that is also an isomorphismIsomorphism

In mathematics, an isomorphism is a bijective map f such that both f and its inverse f −1 are homomo...
is called an automorphismAutomorphism Overview

In mathematics, an automorphism is an isomorphism from a mathematical object to itself....
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 unitsUnit (ring theory)

In mathematics, a unit in a ring R is an invertible element of R, i.e....
in the ring End(V).

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

In mathematics, an isomorphism is a bijective map f such that both f and its inverse f −1 are homomo...
to the associative algebraAssociative algebra

In mathematics, an associative algebra is a vector space which also allows the multiplication of vectors in a distributive ...
of all n by n matrices with entries in K. The automorphism group of V is isomorphicGroup isomorphism Overview

In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between t...
to the general linear groupGeneral linear group

In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with ...
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 kernelKernel (linear operator)

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

In mathematics, image is a part of the set theoretic notion of function. ...
or rangeRange (mathematics)

In mathematics, the range of a function is the set of all "output" values produced by that function....
of f by
ker(f) is a subspaceLinear subspace

The concept of a linear subspace is important in linear algebra and related fields of mathematics....
of V and im(f) is a subspace of W. The following dimensionDimension

In common usage, a dimension is a parameter or measurement required to define the characteristics of an object—i.e....
formula, known as the rank-nullity theoremRank-nullity theorem

In mathematics, the rank-nullity theorem of linear algebra, in its simplest form, relates the rank and the nullity of a matr...
, 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 nullityNull space

In mathematics, the null space of an operator A'' is the set of all operands v' which solve the equation Av''...
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 monomorphismMonomorphism

In the context of abstract algebra or universal algebra, a monomorphism is simply an injective homomorphism....
if any of the following equivalent conditions are true:

T is one-to-one as a map of setsSets Overview

Sets may refer to:*The plural of set...
.
ker T = 0
T is monicMonic

In mathematics, monic can refer to...
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 mapIdentity function Summary

In mathematics, an identity function, also called identity map or identity transformation, is a function which d...
on V.

T is said to be surjective or an epimorphismFacts About Epimorphism

In category theory an epimorphism is a morphism f : X → Y which is "right-cancellable" in the following se...
if any of the following equivalent conditions are true:

T is onto as a map of setsSets

Sets may refer to:*The plural of set...
.
cokerCokernel

In mathematics, the cokernel of a morphism f : X → Y is an object Q and a morphism q : Y →...
T = 0
T is epicEpimorphism Summary

In category theory an epimorphism is a morphism f : X → Y which is "right-cancellable" in the following se...
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 mapIdentity function

In mathematics, an identity function, also called identity map or identity transformation, is a function which d...
on V.

T is said to be an isomorphismIsomorphism Overview

In mathematics, an isomorphism is a bijective map f such that both f and its inverse f −1 are homomo...
if it is both left- and right-invertible. This is equivalent to T being both one-to-one and onto (a bijectionFacts About Bijection

In mathematics, a function f from a set X to a set Y is said to be bijective if for every y in Y there i...
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 nilpotentNilpotent

In mathematics, an element x'' of a ring R'' is called nilpotent if there exists some positive integer n'' such that '...
.
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 spaceTopological vector space

In mathematics a topological vector space is one of the basic structures investigated in functional analysis....
s, for example normed spaces, may also be continuousContinuous function (topology)

In topology and related areas of mathematics a continuous function is a morphism between topological spaces....
and therefore be a continuous linear operatorContinuous linear operator

In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is ...
. On a normed space, a linear operator is continuous if and only if it is boundedBounded operator

In functional analysis, a bounded linear operator is a linear transformation L between normed vector spaces X'' and ...
, 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).
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 graphicsComputer graphics

Computer graphics is the field of visual computing, where one utilizes computers both to generate visual synthetically and...
, where the translation, rotation and scaling of 2D or 3D objects is performed by the use of a transformation matrixTransformation matrix

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.
See also
Linear equationLinear equation

A linear equation is an equation involving only the sum of constants or products of constants and the first power of a vari...
Antilinear mapAntilinear map

In mathematics, a mapping f : V → W from a complex vector space to another is said to be antilinear if...
Transformation matrixTransformation matrix

In linear algebra, linear transformations can be represented by matrices....
Continuous linear operatorFacts About Continuous linear operator

In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is ...

Neural networkNeural network

A neural network is a system of interconnecting neurons in a network working together to produce an output function....
Computer graphicsComputer graphics

Computer graphics is the field of visual computing, where one utilizes computers both to generate visual synthetically and...