Linear transformation
Encyclopedia
In mathematics
, a linear map, linear mapping, linear transformation, or linear operator (in some contexts also called linear function
) is a function
between two vector space
s that preserves the operations of vector addition and scalar
multiplication. As a result, it always maps straight lines to straight lines or 0. The expression "linear operator" is commonly used for linear maps from a vector space to itself (i.e., endomorphisms). Sometimes the definition of a linear function
coincides with that of a linear map, while in analytic geometry
it does not.
In the language of abstract algebra
, a linear map is a homomorphism
of vector spaces. In the language of category theory
it is a morphism
in the category of vector spaces
over a given 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 α in K, the following two conditions are satisfied:
This is equivalent to requiring that for any vectors x_{1}, ..., x_{m} ∈ V and scalars a_{1}, ..., a_{m} ∈ K, the following 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 Klinear maps. For example, the conjugation
of complex numbers is an Rlinear map C → C, but it is not Clinear.
A linear map from V to K (with K viewed as a vector space over itself) is called a linear functional
.
; this is useful because it allows concrete calculations. Conversely, matrices yield examples of linear maps: if A is a real mbyn matrix, then the rule
f(x) = Ax describes a linear map R^{n} → R^{m} (see Euclidean 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 mbyn 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 nby1 matrix C, we have MC = the mby1 matrix whose ith 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.
al space R^{2} linear maps are described by 2 × 2 real matrices. These are some examples:
of all vector spaces over a given field K, together with Klinear maps as morphism
s, forms a category
.
The inverse
of a linear map, when defined, is again a linear map.
If f_{1} : V → W and f_{2} : V → W are linear, then so is their sum f_{1} + f_{2} (which is defined by (f_{1} + f_{2})(x) = f_{1}(x) + f_{2}(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.
Given again the finitedimensional 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.
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.
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).
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.
or range
of f by
ker(f) is a subspace
of V and im(f) is a subspace of W. The following dimension
formula is known as the ranknullity theorem
:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a linear map, linear mapping, linear transformation, or linear operator (in some contexts also called linear function
Linear function
In mathematics, the term linear function can refer to either of two different but related concepts:* a firstdegree polynomial function of one variable;* a map between two vector spaces that preserves vector addition and scalar multiplication....
) is a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
between two vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
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. As a result, it always maps straight lines to straight lines or 0. The expression "linear operator" is commonly used for linear maps from a vector space to itself (i.e., endomorphisms). Sometimes the definition of a linear function
Linear function
In mathematics, the term linear function can refer to either of two different but related concepts:* a firstdegree polynomial function of one variable;* a map between two vector spaces that preserves vector addition and scalar multiplication....
coincides with that of a linear map, while in analytic geometry
Analytic geometry
Analytic geometry, or analytical geometry has two different meanings in mathematics. The modern and advanced meaning refers to the geometry of analytic varieties...
it does not.
In the language of abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, a linear map is a homomorphism
Homomorphism
In abstract algebra, a homomorphism is a structurepreserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape". Definition :The definition of homomorphism depends on the type of algebraic structure under...
of vector spaces. In the language of category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
it is a morphism
Morphism
In mathematics, a morphism is an abstraction derived from structurepreserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
in the category of vector spaces
Category of vector spaces
In mathematics, especially category theory, the category KVect has all vector spaces over a fixed field K as objects and Klinear transformations as morphisms...
over a given field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it 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 fieldField (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it 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 α in K, the following two conditions are satisfied:
additivity Additive function In mathematics the term additive function has two different definitions, depending on the specific field of application.In algebra an additive function is a function that preserves the addition operation:for any two elements x and y in the domain. For example, any linear map is additive... 

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. More precisely, if is a function between two vector spaces over a field F, and k is an integer, then... . 
This is equivalent to requiring that for any vectors x_{1}, ..., x_{m} ∈ V and scalars a_{1}, ..., a_{m} ∈ K, the following 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 Klinear maps. For example, the conjugation
Complex conjugate
In mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...
of complex numbers is an Rlinear map C → C, but it is not Clinear.
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 linear functional or linear form is a linear map from a vector space to its field of scalars. In Rn, if vectors are represented as column vectors, then linear functionals are represented as row vectors, and their action on vectors is given by the dot product, or the...
.
Examples
 The identity mapIdentity functionIn 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.  The map , where c is a constant, is linear.
 For real numbers, the map is not linear.
 For real numbers, the map is not linear (but is an affine transformationAffine transformationIn geometry, an affine transformation or affine map or an affinity is a transformation which preserves straight lines. It is the most general class of transformations with this property...
, and also a linear functionLinear functionIn mathematics, the term linear function can refer to either of two different but related concepts:* a firstdegree polynomial function of one variable;* a map between two vector spaces that preserves vector addition and scalar multiplication....
, as defined in analytic geometryAnalytic geometryAnalytic geometry, or analytical geometry has two different meanings in mathematics. The modern and advanced meaning refers to the geometry of analytic varieties...
.)  If A is a real m × n matrixMatrix (mathematics)In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
, then A defines a linear map from R^{n} to R^{m} by sending the column vector x ∈ R^{n} to the column vector Ax ∈ R^{m}. Conversely, any linear map between finitedimensional vector spaces can be represented in this manner; see the following section.  The (definite) integralIntegralIntegration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
is a linear map from the space of all realvalued integrable functions on some intervalInterval (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. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...
to R  The (indefinite) integralIntegralIntegration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
(or antiderivativeAntiderivativeIn calculus, an "antiderivative", antiderivative, primitive integral or indefinite integralof a function f is a function F whose derivative is equal to f, i.e., F ′ = f...
) is not considered a linear transformation, as the use of a constant of integration results in an infinite number of outputs per input.  DifferentiationDerivativeIn 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 one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
is a linear map from the space of all differentiable functions to the space of all functions.  If V and W are finitedimensional vector spaces over a field F, then functions that send linear maps f : V → W to dim_{F}(W)bydim_{F}(V) matrices in the way described in the sequel are themselves linear maps.
 The expected valueExpected valueIn probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
of a random variable X is linear, as , but the variance of a random variable is not linear, as it violates the second condition, homogeneity of degree 1: .
Matrices
If V and W are finitedimensional, 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 array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
; this is useful because it allows concrete calculations. Conversely, matrices yield examples of linear maps: if A is a real mbyn matrix, then the rule
f(x) = Ax describes a linear map R^{n} → R^{m} (see Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and threedimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
).
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 mbyn 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 nby1 matrix C, we have MC = the mby1 matrix whose ith 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
In twodimensionDimension
In physics and mathematics, the dimension of a 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 because only one coordinate is needed to specify a point on it...
al space R^{2} linear maps are described by 2 × 2 real matrices. These are some examples:
 rotationRotation (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. A rotation is different from a translation, which has no fixed points, and from a reflection, which "flips" the bodies it is transforming...
by 90 degrees counterclockwise:  rotationRotation (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. A rotation is different from a translation, which has no fixed points, and from a reflection, which "flips" the bodies it is transforming...
by θ degrees counterclockwise:  reflectionReflection (mathematics)In mathematics, a reflection is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as set of fixed points; this set is called the axis or plane of reflection. The image of a figure by a reflection is its mirror image in the axis or plane of reflection...
against the x axis:  reflectionReflection (mathematics)In mathematics, a reflection is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as set of fixed points; this set is called the axis or plane of reflection. The image of a figure by a reflection is its mirror image in the axis or plane of reflection...
against the y axis:  scalingScaling (geometry)In Euclidean geometry, uniform scaling is a linear transformation that enlarges or shrinks objects by a scale factor that is the same in all directions. The result of uniform scaling is similar to the original...
by 2 in all directions:  horizontal shear mapping:
 squeeze mappingSqueeze mappingIn 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.For a fixed positive real number r, the mapping →...
:  projectionProjection (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. It leaves its image unchanged....
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 their composition g o f : V → Z. It follows from this that the classClass (set theory)
In set theory and its applications throughout mathematics, a class is a collection of sets which can be unambiguously defined by a property that all its members share. The precise definition of "class" depends on foundational context...
of all vector spaces over a given field K, together with Klinear maps as morphism
Morphism
In mathematics, a morphism is an abstraction derived from structurepreserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
s, forms a category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
.
The inverse
Inverse function
In mathematics, an inverse function is a function that undoes another function: If an input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa. i.e., ƒ=y, and g=x...
of a linear map, when defined, is again a linear map.
If f_{1} : V → W and f_{2} : V → W are linear, then so is their sum f_{1} + f_{2} (which is defined by (f_{1} + f_{2})(x) = f_{1}(x) + f_{2}(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 A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R...
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 finitedimensional case, if bases have been chosen, then the composition of linear maps corresponds to the matrix multiplication
Matrix multiplication
In mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. If A is an nbym matrix and B is an mbyp matrix, the result AB of their multiplication is an nbyp matrix defined only if the number of columns m of the left matrix A is the...
, the addition of linear maps corresponds to the matrix addition
Matrix addition
In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together. However, there are other operations which could also be considered as a kind of addition for matrices, the direct sum and the Kronecker sum....
, 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. For example, an endomorphism of a vector space V is a linear map ƒ: V → V, and an endomorphism of a group G is a group homomorphism ƒ: G → G. In general, we can talk about...
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 A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R...
with identity element over the field K (and in particular a ring). The multiplicative identity element of this algebra 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...
id : V → V.
An endomorphism of V that is also an isomorphism
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
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 mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the 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
Unit (ring theory)
In mathematics, an invertible element or a unit in a ring R refers to any element u that has an inverse element in the multiplicative monoid of R, i.e. such element v that...
in the ring End(V).
If V has finite dimension n, then End(V) is isomorphic
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
to the associative algebra
Associative algebra
In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R...
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 groups that sets up a onetoone correspondence between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, then the groups are called isomorphic...
to the general linear group
General linear group
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. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...
GL(n, K) of all n by n invertible matrices with entries in K.
Kernel, image and the ranknullity theorem
If f : V → W is linear, we define the kernel and the imageImage (mathematics)
In mathematics, an image is the subset of a function's codomain which is the output of the function on a subset of its domain. Precisely, evaluating the function at each element of a subset X of the domain produces a set called the image of X under or through the function...
or range
Range (mathematics)
In mathematics, the range of a function refers to either the codomain or the image of the function, depending upon usage. This ambiguity is illustrated by the function f that maps real numbers to real numbers with f = x^2. Some books say that range of this function is its codomain, the set of all...
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 physics and mathematics, the dimension of a 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 because only one coordinate is needed to specify a point on it...
formula is known as the ranknullity theorem
Ranknullity 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. Specifically, if A is an mbyn matrix over some field, thenThis applies to linear maps as well...
:

The number dim(im(f)) is also called the rank of f and written as rank(f), or sometimes, ρ(f); the number dim(ker(f)) is called the nullity of f and written as null(f) or ν(f). If V and W are finitedimensional, 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 is the cokernelCokernelIn 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....
, which is defined as
This is the dual notion to the kernel: just as the kernel is a subspace of the domain, the cokernel is a quotient spaceQuotient space (linear algebra)In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by "collapsing" N to zero. The space obtained is called a quotient space and is denoted V/N ....
of the target.
Formally, one has the exact sequenceExact sequenceAn exact sequence is a concept in mathematics, especially in homological algebra and other applications of abelian category theory, as well as in differential geometry and group theory...
These can be interpreted thus: given a linear equation to solve,
 the kernel is the space of solutions to the homogeneous equation and its dimension is the number of degrees of freedom in a solution, if it exists;
 the cokernel is the space of constraints that must be satisfied if the equation is to have a solution, and its dimension is the number of constraints that must be satisfied for the equation to have a solution.
The dimension of the cokernel 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 is the dimension of the target space minus the dimension of the image.
As a simple example, consider the map given by
Then for an equation to have a solution, we must have (one constraint), and in that case the solution space is or equivalently stated, (one degree of freedom). The kernel may be expressed as the subspace the value of x is the freedom in a solution – while the cokernel may be expressed via the map given a vector the value of a is the obstruction to there being a solution.
An example illustrating the infinitedimensional case is afforded by the map with and for . 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 cokernel 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 cokernel ( ), but in the infinitedimensional case it cannot be inferred that the kernel and the cokernel of an endomorphismEndomorphismIn 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 ƒ: V → V, and an endomorphism of a group G is a group homomorphism ƒ: G → G. In general, we can talk about...
have the same dimension (). The reverse situation obtains for the map with . Its image is the entire target space, and hence its cokernel has dimension 0, but since it maps all sequences in which only the first element is nonzero to the zero sequence, its kernel has dimension 1.
Index
For a linear operator with finitedimensional kernel and cokernel, one may define index as:
namely the degrees of freedom minus the number of constraints.
For a transformation between finitedimensional vector spaces, this is just the difference by ranknullity. 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 comes of its own in infinite dimensions: it is how homologyHomology (mathematics)In mathematics , homology is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group...
is defined, which is a central theory in algebra and algebraic topologyAlgebraic topologyAlgebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
; the index of an operator is precisely the Euler characteristicEuler characteristicIn mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...
of the 2term complex
In operator theoryOperator theoryIn mathematics, operator theory is the branch of functional analysis that focuses on bounded linear operators, but which includes closed operators and nonlinear operators.Operator theory also includes the study of algebras of operators....
, the index of FredholmFredholmFredholm is surname of:* Ivar Fredholm , Swedish mathematician** Fredholm alternative, in mathematics** Fredholm determinant, in mathematics** Fredholm integral equation, in mathematics** Fredholm kernel, in mathematics...
operators is an object of study, with a major result being the Atiyah–Singer index theoremAtiyah–Singer index theoremIn differential geometry, the Atiyah–Singer index theorem, proved by , states that for an elliptic differential operator on a compact manifold, the analytical index is equal to the topological index...
.
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 monomorphismMonomorphismIn the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation X \hookrightarrow Y....
if any of the following equivalent conditions are true: T is onetoone as a map of sets.
 kerT = {0_{V}}
 T is monic or leftcancellable, 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 leftinvertible, which is to say there exists a linear map S:W → V such that ST is the identity mapIdentity functionIn 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 epimorphismEpimorphismIn category theory, an epimorphism is a morphism f : X → Y which is rightcancellative in the sense that, for all morphisms ,...
if any of the following equivalent conditions are true: T is onto as a map of sets.
 cokerCokernelIn 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_{W}}  T is epicEpimorphismIn category theory, an epimorphism is a morphism f : X → Y which is rightcancellative in the sense that, for all morphisms ,...
or rightcancellable, 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 rightinvertible, which is to say there exists a linear map S:W → V such that TS is the identity mapIdentity functionIn 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 W.
 T is said to be an isomorphismIsomorphismIn abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
if it is both left and rightinvertible. This is equivalent to T being both onetoone and onto (a bijectionBijectionA bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...
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 nth iterate of T, T^{n}, is identically zero, then T is said to be nilpotentNilpotentIn 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; see scalar matrix.
 If, for some positive integer n, the nth iterate of T, T^{n}, is identically zero, then T is said to be nilpotent
Change of basis
Given a linear map whose matrix is A, in the basis B of the space in transforms vectors coordinates [u] as [v]=A⋅[u]. As vectors change with the inverse of B, its inverse transformation is [v]=B⋅[v'].
Substituting this in the first expression:
Therefore the matrix in the new basis is , being B the matrix of the given basis.
Therefore linear maps are said to be 1co 1contra variant objects, or type (1, 1) tensorTensorTensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multidimensional array of...
s.
Continuity
A linear transformation between topological vector spaceTopological vector spaceIn mathematics, a topological vector space is one of the basic structures investigated in functional analysis...
s, for example normed spaces, may be continuous. If its domain and codomain are the same, it will then be a continuous linear operatorContinuous linear operatorIn functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces....
. A linear operator on a normed linear space is continuous if and only if it is boundedBounded operatorIn functional analysis, a branch of mathematics, 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 by the same number, over all nonzero vectors v in X...
, for example, when the domain is finitedimensional. An infinitedimensional 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). Differentiation is not a linear operator: its codomain is larger than its domain, because the derivative of a smooth function need not be smooth.
Applications
A specific application of linear maps is for geometric transformations, such as those performed in computer graphicsComputer graphicsComputer graphics are graphics created using computers and, more generally, the representation and manipulation of image data by a computer with help from specialized software and hardware....
, where the translation, rotation and scaling of 2D or 3D objects is performed by the use of a transformation matrix.
Another application of these transformations is in compiler optimizations of nestedloop code, and in parallelizing compiler techniques.
See also
 Affine transformationAffine transformationIn geometry, an affine transformation or affine map or an affinity is a transformation which preserves straight lines. It is the most general class of transformations with this property...
 Linear equationLinear equationA linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable....
 Bounded operatorBounded operatorIn functional analysis, a branch of mathematics, 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 by the same number, over all nonzero vectors v in X...
 Antilinear map
 Semilinear transformationSemilinear transformationIn linear algebra, particularly projective geometry, a semilinear transformation between vector spaces V and W over a field K is a function that is a linear transformation "up to a twist", hence semilinear, where "twist" means "field automorphism of K"...
 Continuous linear operatorContinuous linear operatorIn functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces....
 wikibooks:Linear Algebra/Linear Transformations
 Neural networkNeural networkThe term neural network was traditionally used to refer to a network or circuit of biological neurons. The modern usage of the term often refers to artificial neural networks, which are composed of artificial neurons or nodes...
 Computer graphicsComputer graphicsComputer graphics are graphics created using computers and, more generally, the representation and manipulation of image data by a computer with help from specialized software and hardware....