linear transformation

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OR:

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, and more specifically in
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices ...
, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping $V \to W$ between two
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called ''vector (mathematics and physics), vectors'', may be Vector addition, added together and Scalar multiplication, mu ...
s that preserves the operations of
vector addition In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors ...
and scalar multiplication. The same names and the same definition are also used for the more general case of modules over a ring; see
Module homomorphism In algebra, a module homomorphism is a function between modules that preserves the module structures. Explicitly, if ''M'' and ''N'' are left modules over a ring ''R'', then a function f: M \to N is called an ''R''-''module homomorphism'' or an '' ...
. If a linear map is a
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
then it is called a . In the case where $V = W$, a linear map is called a (linear) ''
endomorphism In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a g ...
''. Sometimes the term refers to this case, but the term "linear operator" can have different meanings for different conventions: for example, it can be used to emphasize that $V$ and $W$ are real vector spaces (not necessarily with $V = W$), or it can be used to emphasize that $V$ is a
function space In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a ve ...
, which is a common convention in
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined ...
. Sometimes the term ''
linear function In mathematics, the term linear function refers to two distinct but related notions: * In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. For dist ...
'' has the same meaning as ''linear map'', while in
analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (384 ...
it does not. A linear map from ''V'' to ''W'' always maps the origin of ''V'' to the origin of ''W''. Moreover, it maps
linear subspace In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, ...
s in ''V'' onto linear subspaces in ''W'' (possibly of a lower
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordin ...
); for example, it maps a plane through the origin in ''V'' to either a plane through the origin in ''W'', a
line Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Art ...
through the origin in ''W'', or just the origin in ''W''. Linear maps can often be represented as
matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
, and simple examples include rotation and reflection linear transformations. In the language of
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cat ...
, linear maps are the
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphism ...
s of vector spaces.

# Definition and first consequences

Let $V$ and $W$ be vector spaces over the same field $K$. A function $f: V \to W$ is said to be a ''linear map'' if for any two vectors $\mathbf, \mathbf \in V$ and any scalar $c \in K$ the following two conditions are satisfied: * Additivity / operation of addition $f(\mathbf + \mathbf) = f(\mathbf) + f(\mathbf)$ *
Homogeneity Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, ...
of degree 1 / operation of scalar multiplication $f(c \mathbf) = c f(\mathbf)$ Thus, a linear map is said to be ''operation preserving''. In other words, it does not matter whether the linear map is applied before (the right hand sides of the above examples) or after (the left hand sides of the examples) the operations of addition and scalar multiplication. By the associativity of the addition operation denoted as +, for any vectors $\mathbf_1, \ldots, \mathbf_n \in V$ and scalars $c_1, \ldots, c_n \in K,$ the following equality holds: $f(c_1 \mathbf_1 + \cdots + c_n \mathbf_n) = c_1 f(\mathbf_1) + \cdots + c_n f(\mathbf_n).$ Thus a linear map is one which preserves linear combinations. Denoting the zero elements of the vector spaces $V$ and $W$ by $\mathbf_V$ and $\mathbf_W$ respectively, it follows that $f(\mathbf_V) = \mathbf_W.$ Let $c = 0$ and $\mathbf \in V$ in the equation for homogeneity of degree 1: $f(\mathbf_V) = f(0\mathbf) = 0f(\mathbf) = \mathbf_W.$ A linear map $V \to K$ with $K$ viewed as a one-dimensional vector space over itself is called a
linear functional In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , ...
. These statements generalize to any left-module $_R M$ over a ring $R$ without modification, and to any right-module upon reversing of the scalar multiplication.

# Examples

* A prototypical example that gives linear maps their name is a function $f: \mathbb \to \mathbb: x \mapsto cx$, of which the graph is a line through the origin. * More generally, any homothety $\mathbf \mapsto c\mathbf$ where $c$ centered in the origin of a vector space is a linear map. * The zero map $\mathbf x \mapsto \mathbf 0$ between two vector spaces (over the same field) is linear. * The identity map on any module is a linear operator. * For real numbers, the map $x \mapsto x^2$ is not linear. * For real numbers, the map $x \mapsto x + 1$ is not linear (but is an
affine transformation In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generall ...
). * If $A$ is a $m \times n$
real matrix In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, \beg ...
, then $A$ defines a linear map from $\R^n$ to $\R^m$ by sending a
column vector In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example, \boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end. Similarly, a row vector is a 1 \times n matrix for some n, ...
$\mathbf x \in \R^n$ to the column vector $A \mathbf x \in \R^m$. Conversely, any linear map between
finite-dimensional In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to dist ...
vector spaces can be represented in this manner; see the , below. * If $f: V \to W$ is an
isometry In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
between real
normed space In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" ...
s such that $f(0) = 0$ then $f$ is a linear map. This result is not necessarily true for complex normed space. * Differentiation defines a linear map from the space of all differentiable functions to the space of all functions. It also defines a linear operator on the space of all
smooth function In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
s (a linear operator is a linear
endomorphism In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a g ...
, that is, a linear map with the same domain and
codomain In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either the ...
). An example is $\frac \left( c_1 f_1(x) + c_2 f_2(x) + \cdots + c_n f_n(x) \right) = c_1 \frac + c_2 \frac + \cdots + c_n \frac.$ * A definite
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
over some interval is a linear map from the space of all real-valued integrable functions on to $\R$. For example, * An indefinite
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
(or
antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolically ...
) with a fixed integration starting point defines a linear map from the space of all real-valued integrable functions on $\R$ to the space of all real-valued, differentiable functions on $\R$. Without a fixed starting point, the antiderivative maps to the quotient space of the differentiable functions by the linear space of constant functions. * If $V$ and $W$ are finite-dimensional vector spaces over a field , of respective dimensions and , then the function that maps linear maps $f: V \to W$ to matrices in the way described in (below) is a linear map, and even a
linear isomorphism In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that p ...
. * The
expected value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
of a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
(which is in fact a function, and as such a element of a vector space) is linear, as for random variables $X$ and $Y$ we have
variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
of a random variable is not linear. File:Streckung eines Vektors.gif, The function $f:\R^2 \to \R^2$ with $f(x, y) = (2x, y)$ is a linear map. This function scales the $x$ component of a vector by the factor $2$. File:Streckung der Summe zweier Vektoren.gif, The function $f(x, y) = (2x, y)$ is additive: It doesn't matter whether vectors are first added and then mapped or whether they are mapped and finally added: $f(\mathbf a + \mathbf b) = f(\mathbf a) + f(\mathbf b)$ File:Streckung homogenitaet Version 3.gif, The function $f(x, y) = (2x, y)$ is homogeneous: It doesn't matter whether a vector is first scaled and then mapped or first mapped and then scaled: $f(\lambda \mathbf a) = \lambda f(\mathbf a)$

## Linear extensions

Often, a linear map is constructed by defining it on a subset of a vector space and then to the
linear span In mathematics, the linear span (also called the linear hull or just span) of a set of vectors (from a vector space), denoted , pp. 29-30, §§ 2.5, 2.8 is defined as the set of all linear combinations of the vectors in . It can be characteriz ...
of the domain. A ' of a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orien ...
$f$ is an extension of $f$ to some
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called ''vector (mathematics and physics), vectors'', may be Vector addition, added together and Scalar multiplication, mu ...
that is a linear map. Suppose $X$ and $Y$ are vector spaces and $f : S \to Y$ is a function defined on some subset $S \subseteq X.$ Then $f$ can be extended to a linear map $F : \operatorname S \to Y$ if and only if whenever $n > 0$ is an integer, $c_1, \ldots, c_n$ are scalars, and $s_1, \ldots, s_n \in S$ are vectors such that $0 = c_1 s_1 + \cdots + c_n s_n,$ then necessarily $0 = c_1 f\left\left(s_1\right\right) + \cdots + c_n f\left\left(s_n\right\right).$ If a linear extension of $f : S \to Y$ exists then the linear extension $F : \operatorname S \to Y$ is unique and $F\left(c_1 s_1 + \cdots c_n s_n\right) = c_1 f\left(s_1\right) + \cdots + c_n f\left(s_n\right)$ holds for all $n, c_1, \ldots, c_n,$ and $s_1, \ldots, s_n$ as above. If $S$ is linearly independent then every function $f : S \to Y$ into any vector space has a linear extension to a (linear) map $\;\operatorname S \to Y$ (the converse is also true). For example, if $X = \R^2$ and $Y = \R$ then the assignment $\left(1, 0\right) \to -1$ and $\left(0, 1\right) \to 2$ can be linearly extended from the linearly independent set of vectors $S := \$ to a linear map on $\operatorname\ = \R^2.$ The unique linear extension $F : \R^2 \to \R$ is the map that sends $\left(x, y\right) = x \left(1, 0\right) + y \left(0, 1\right) \in \R^2$ to $F(x, y) = x (-1) + y (2) = - x + 2 y.$ Every (scalar-valued)
linear functional In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , ...
$f$ defined on a vector subspace of a real or complex vector space $X$ has a linear extension to all of $X.$ Indeed, the Hahn–Banach dominated extension theorem even guarantees that when this linear functional $f$ is dominated by some given seminorm $p : X \to \R$ (meaning that $, f\left(m\right), \leq p\left(m\right)$ holds for all $m$ in the domain of $f$) then there exists a linear extension to $X$ that is also dominated by $p.$

# Matrices

If $V$ and $W$ are
finite-dimensional In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to dist ...
vector spaces and a basis is defined for each vector space, then every linear map from $V$ to $W$ can be represented by a matrix. This is useful because it allows concrete calculations. Matrices yield examples of linear maps: if $A$ is a real $m \times n$ matrix, then $f\left(\mathbf x\right) = A \mathbf x$ describes a linear map $\R^n \to \R^m$ (see
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
). Let $\$ be a basis for $V$. Then every vector $\mathbf \in V$ is uniquely determined by the coefficients $c_1, \ldots , c_n$ in the field $\R$: $\mathbf = c_1 \mathbf_1 + \cdots + c_n \mathbf _n.$ If $f: V \to W$ is a linear map, $f(\mathbf) = f(c_1 \mathbf_1 + \cdots + c_n \mathbf_n) = c_1 f(\mathbf_1) + \cdots + c_n f\left(\mathbf_n\right),$ which implies that the function ''f'' is entirely determined by the vectors $f\left(\mathbf _1\right), \ldots , f\left(\mathbf _n\right)$. Now let $\$ be a basis for $W$. Then we can represent each vector $f\left(\mathbf _j\right)$ as $f\left(\mathbf_j\right) = a_ \mathbf_1 + \cdots + a_ \mathbf_m.$ Thus, the function $f$ is entirely determined by the values of $a_$. If we put these values into an $m \times n$ matrix $M$, then we can conveniently use it to compute the vector output of $f$ for any vector in $V$. To get $M$, every column $j$ of $M$ is a vector $\begin a_ \\ \vdots \\ a_ \end$ corresponding to $f\left(\mathbf _j\right)$ as defined above. To define it more clearly, for some column $j$ that corresponds to the mapping $f\left(\mathbf _j\right)$, $\mathbf = \begin \ \cdots & a_ & \cdots\ \\ & \vdots & \\ & a_ & \end$ where $M$ is the matrix of $f$. In other words, every column $j = 1, \ldots, n$ has a corresponding vector $f\left(\mathbf _j\right)$ whose coordinates $a_, \cdots, a_$ are the elements of column $j$. A single linear map may be represented by many matrices. This is because the values of the elements of a matrix depend on the bases chosen. The matrices of a linear transformation can be represented visually: # Matrix for $T$ relative to $B$: $A$ # Matrix for $T$ relative to $B'$: $A'$ # Transition matrix from $B'$ to $B$: $P$ # Transition matrix from $B$ to $B'$: $P^$ Such that starting in the bottom left corner and looking for the bottom right corner , one would left-multiply—that is, . The equivalent method would be the "longer" method going clockwise from the same point such that is left-multiplied with $P^AP$, or .

## Examples in two dimensions

In two-
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordin ...
al space R2 linear maps are described by 2 × 2
matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
. These are some examples: *
rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
** by 90 degrees counterclockwise: $\mathbf = \begin 0 & -1\\ 1 & 0\end$ ** by an angle ''θ'' counterclockwise: $\mathbf = \begin \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end$ * reflection ** through the ''x'' axis: $\mathbf = \begin 1 & 0\\ 0 & -1\end$ ** through the ''y'' axis: $\mathbf = \begin-1 & 0\\ 0 & 1\end$ ** through a line making an angle ''θ'' with the origin: $\mathbf = \begin\cos2\theta & \sin2\theta \\ \sin2\theta & -\cos2\theta \end$ *
scaling Scaling may refer to: Science and technology Mathematics and physics * Scaling (geometry), a linear transformation that enlarges or diminishes objects * Scale invariance, a feature of objects or laws that do not change if scales of length, energ ...
by 2 in all directions: $\mathbf = \begin 2 & 0\\ 0 & 2\end = 2\mathbf$ * horizontal shear mapping: $\mathbf = \begin 1 & m\\ 0 & 1\end$ *
squeeze mapping In linear algebra, a squeeze mapping, also called a squeeze transformation, is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is ''not'' a rotation or shear mapping. For a fixed positive real number , ...
: $\mathbf = \begin k & 0\\ 0 & \frac\end$ * projection onto the ''y'' axis: $\mathbf = \begin 0 & 0\\ 0 & 1\end.$

# Vector space of linear maps

The composition of linear maps is linear: if $f: V \to W$ and $g: W \to Z$ are linear, then so is their composition $g \circ f: V \to Z$. It follows from this that the class of all vector spaces over a given field ''K'', together with ''K''-linear maps as
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphism ...
s, forms a
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) * ...
. The inverse of a linear map, when defined, is again a linear map. If $f_1: V \to W$ and $f_2: V \to W$ are linear, then so is their
pointwise In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined ...
sum $f_1 + f_2$, which is defined by $\left(f_1 + f_2\right)\left(\mathbf x\right) = f_1\left(\mathbf x\right) + f_2\left(\mathbf x\right)$. If $f: V \to W$ is linear and $\alpha$ is an element of the ground field $K$, then the map $\alpha f$, defined by $(\alpha f)(\mathbf x) = \alpha (f(\mathbf x))$, is also linear. Thus the set $\mathcal(V, W)$ of linear maps from $V$ to $W$ itself forms a vector space over $K$, sometimes denoted $\operatorname(V, W)$. Furthermore, in the case that $V = W$, this vector space, denoted $\operatorname(V)$, is an
associative algebra In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplicat ...
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 In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the ...
, the addition of linear maps corresponds to the
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 addition for matrices, such as the direct sum and the Kroneck ...
, and the multiplication of linear maps with scalars corresponds to the multiplication of matrices with scalars.

## Endomorphisms and automorphisms

A linear transformation $f : V \to V$ is an
endomorphism In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a g ...
of $V$; the set of all such endomorphisms $\operatorname(V)$ together with addition, composition and scalar multiplication as defined above forms an
associative algebra In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplicat ...
with identity element over the field $K$ (and in particular a ring). The multiplicative identity element of this algebra is the identity map $\operatorname: V \to V$. An endomorphism of $V$ that is also an
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
is called an automorphism 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 $\operatorname(V)$ or $\operatorname(V)$. Since the automorphisms are precisely those endomorphisms which possess inverses under composition, $\operatorname(V)$ is the group of units in the ring $\operatorname(V)$. If $V$ has finite dimension $n$, then $\operatorname(V)$ is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to the
associative algebra In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplicat ...
of all $n \times n$ matrices with entries in $K$. The automorphism group of $V$ is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to the
general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
$\operatorname(n, K)$ of all $n \times n$ invertible matrices with entries in $K$.

# Kernel, image and the rank–nullity theorem

If $f: V \to W$ is linear, we define the
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learni ...
and the
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimension ...
or
range Range may refer to: Geography * Range (geographic), a chain of hills or mountains; a somewhat linear, complex mountainous or hilly area (cordillera, sierra) ** Mountain range, a group of mountains bordered by lowlands * Range, a term used to ...
of $f$ by $\begin \ker(f) &= \ \\ \operatorname(f) &= \ \end$ $\ker(f)$ is a subspace of $V$ and $\operatorname(f)$ is a subspace of $W$. The following
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordin ...
formula is known as the
rank–nullity theorem The rank–nullity theorem is a theorem in linear algebra, which asserts that the dimension of the domain of a linear map is the sum of its rank (the dimension of its image) and its ''nullity'' (the dimension of its kernel). p. 70, §2.1, Th ...
: $\dim(\ker( f )) + \dim(\operatorname( f )) = \dim( V ).$ The number $\dim(\operatorname(f))$ is also called the
rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * ...
of $f$ and written as $\operatorname(f)$, or sometimes, $\rho(f)$; p. 52, § 2.5.1 p. 90, § 50 the number $\dim(\ker(f))$ is called the nullity of $f$ and written as $\operatorname(f)$ or $\nu(f)$. If $V$ and $W$ are finite-dimensional, bases have been chosen and $f$ is represented by the matrix $A$, then the rank and nullity of $f$ are equal to the rank and nullity of the matrix $A$, respectively.

# Cokernel

A subtler invariant of a linear transformation $f: V \to W$ is the ''co''kernel, which is defined as $\operatorname(f) := W/f(V) = W/\operatorname(f).$ This is the ''dual'' notion to the kernel: just as the kernel is a ''sub''space of the ''domain,'' the co-kernel is a ''quotient'' space of the ''target.'' Formally, one has the
exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the conte ...
$0 \to \ker(f) \to V \to W \to \operatorname(f) \to 0.$ These can be interpreted thus: given a linear equation ''f''(v) = w to solve, * the kernel is the space of ''solutions'' to the ''homogeneous'' equation ''f''(v) = 0, and its dimension is the number of
degrees of freedom Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
in the space of solutions, if it is not empty; * the co-kernel is the space of constraints that the solutions must satisfy, and its dimension is the maximal number of independent constraints. The dimension of the co-kernel and the dimension of the image (the rank) add up to the dimension of the target space. For finite dimensions, this means that the dimension of the quotient space ''W''/''f''(''V'') is the dimension of the target space minus the dimension of the image. As a simple example, consider the map ''f'': R2 → R2, given by ''f''(''x'', ''y'') = (0, ''y''). Then for an equation ''f''(''x'', ''y'') = (''a'', ''b'') to have a solution, we must have ''a'' = 0 (one constraint), and in that case the solution space is (''x'', ''b'') or equivalently stated, (0, ''b'') + (''x'', 0), (one degree of freedom). The kernel may be expressed as the subspace (''x'', 0) < ''V'': the value of ''x'' is the freedom in a solution – while the cokernel may be expressed via the map ''W'' → R, $(a, b) \mapsto (a)$: given a vector (''a'', ''b''), the value of ''a'' is the ''obstruction'' to there being a solution. An example illustrating the infinite-dimensional case is afforded by the map ''f'': R → R, $\left\ \mapsto \left\$ with ''b''1 = 0 and ''b''''n'' + 1 = ''an'' for ''n'' > 0. Its image consists of all sequences with first element 0, and thus its cokernel consists of the classes of sequences with identical first element. Thus, whereas its kernel has dimension 0 (it maps only the zero sequence to the zero sequence), its co-kernel has dimension 1. Since the domain and the target space are the same, the rank and the dimension of the kernel add up to the same sum as the rank and the dimension of the co-kernel ($\aleph_0 + 0 = \aleph_0 + 1$), but in the infinite-dimensional case it cannot be inferred that the kernel and the co-kernel of an
endomorphism In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a g ...
have the same dimension (0 ≠ 1). The reverse situation obtains for the map ''h'': R → R, $\left\ \mapsto \left\$ with ''cn'' = ''a''''n'' + 1. Its image is the entire target space, and hence its co-kernel has dimension 0, but since it maps all sequences in which only the first element is non-zero to the zero sequence, its kernel has dimension 1.

## Index

For a linear operator with finite-dimensional kernel and co-kernel, one may define ''index'' as: $\operatorname(f) := \dim(\ker(f)) - \dim(\operatorname(f)),$ namely the degrees of freedom minus the number of constraints. For a transformation between finite-dimensional vector spaces, this is just the difference dim(''V'') − dim(''W''), by rank–nullity. This gives an indication of how many solutions or how many constraints one has: if mapping from a larger space to a smaller one, the map may be onto, and thus will have degrees of freedom even without constraints. Conversely, if mapping from a smaller space to a larger one, the map cannot be onto, and thus one will have constraints even without degrees of freedom. The index of an operator is precisely the
Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space' ...
of the 2-term complex 0 → ''V'' → ''W'' → 0. In
operator theory In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operator ...
, the index of
Fredholm operator In mathematics, Fredholm operators are certain operators that arise in the Fredholm theory of integral equations. They are named in honour of Erik Ivar Fredholm. By definition, a Fredholm operator is a bounded linear operator ''T'' : ' ...
s is an object of study, with a major result being the Atiyah–Singer index theorem.

# Algebraic classifications of linear transformations

No classification of linear maps could be exhaustive. The following incomplete list enumerates some important classifications that do not require any additional structure on the vector space. Let and denote vector spaces over a field and let be a linear map.

## Monomorphism

is said to be ''
injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...
'' or a ''
monomorphism In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of category theory, a monomorphi ...
'' if any of the following equivalent conditions are true: # is one-to-one as a map of sets. # # # is monic or left-cancellable, which is to say, for any vector space and any pair of linear maps and , the equation implies . # is left-invertible, which is to say there exists a linear map such that is the identity map on .

## Epimorphism

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

## Isomorphism

is said to be an ''
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
'' if it is both left- and right-invertible. This is equivalent to being both one-to-one and onto (a
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
of sets) or also to being both epic and monic, and so being a
bimorphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphism ...
. If is an endomorphism, then: * If, for some positive integer , the -th iterate of , , is identically zero, then is said to be
nilpotent In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the cla ...
. * If , then is said to be
idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
* If , where is some scalar, then is said to be a scaling transformation or scalar multiplication map; see
scalar matrix In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal m ...
.

# Change of basis

Given a linear map which is an
endomorphism In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a g ...
whose matrix is ''A'', in the basis ''B'' of the space it transforms vector coordinates as = ''A'' As vectors change with the inverse of ''B'' (vectors are contravariant) its inverse transformation is = ''B'' ' Substituting this in the first expression
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
s.

# Continuity

A ''linear transformation'' between
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
s, for example
normed space In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" ...
s, may be continuous. If its domain and codomain are the same, it will then be a
continuous linear operator 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. An operator between two normed spaces is a bounded linear ...
. A linear operator on a normed linear space is continuous if and only if it is bounded, for example, when the domain is finite-dimensional. 1.18 Theorem ''Let $\Lambda$ be a linear functional on a topological vector space . Assume $\Lambda \mathbf x \neq 0$ for some $\mathbf x \in X$. Then each of the following four properties implies the other three:'' An infinite-dimensional domain may have discontinuous linear operators. An example of an unbounded, hence discontinuous, linear transformation is differentiation on the space of smooth functions equipped with the supremum norm (a function with small values can have a derivative with large values, while the derivative of 0 is 0). For a specific example, converges to 0, but its derivative does not, so differentiation is not continuous at 0 (and by a variation of this argument, it is not continuous anywhere).

# Applications

A specific application of linear maps is for geometric transformations, such as those performed in
computer graphics Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great dea ...
, where the translation, rotation and scaling of 2D or 3D objects is performed by the use of a transformation matrix. Linear mappings also are used as a mechanism for describing change: for example in calculus correspond to derivatives; or in relativity, used as a device to keep track of the local transformations of reference frames. Another application of these transformations is in compiler optimizations of nested-loop code, and in parallelizing compiler techniques.