Linear map

{{short description|Mathematical function, in linear algebra}} {{redirect|Linear transformation|fractional linear transformations|Möbius transformation}} {{redirect|Linear Operators|the textbook|Linear Operators (book)}} {{distinguish|linear function}} {{more footnotes needed|date=December 2021}} In [[mathematics]], and more specifically in [[linear algebra]], a '''linear map''' (or '''linear mapping''') is a particular kind of [[function (mathematics)|function]] between [[vector space]]s, which respects the basic operations of [[vector addition]] and [[scalar multiplication]]. A standard example of a linear map is an <math>m\times n</math> matrix, which takes vectors in <math>n</math>-dimensions into vectors in <math>m</math>-dimensions in a way that is compatible with addition of vectors, and multiplication of vectors by [[Scalar (mathematics)|scalars]].

A linear map is a [[homomorphism]] of vector spaces.<ref>In the language of [[category theory]], linear maps are the [[morphism]]s of vector spaces. Restricted to the category of finite-dimensional vector spaces, they form a category [[equivalence of categories|equivalent]] to [[category of matrices|the one of matrices]].</ref> Thus, a linear map <math>T:V\to W</math> satisfies {{tmath|1= T(a x + b y) = a Tx + b Ty }}, where <math>a</math> and <math>b</math> are scalars, and <math>x</math> and <math>y</math> are vectors (elements of the vector space {{tmath|V}}). A linear mapping always maps the [[Origin (geometry)|origin]] of <math>V</math> to the origin of {{tmath|W}}, and [[linear subspace]]s of <math>V</math> onto linear subspaces in <math>W</math> (possibly of a lower [[Dimension (vector space)|dimension]]);<ref>{{harvnb|Rudin|1991|page=14}} Here are some properties of linear mappings <math display="inline">\Lambda: X \to Y</math> whose proofs are so easy that we omit them; it is assumed that <math display="inline">A \subset X</math> and <math display="inline">B \subset Y</math>: {{ordered list|<math display="inline">\Lambda 0 = 0.</math>|If {{mvar|A}} is a subspace (or a [[convex set]], or a [[balanced set]]) the same is true of <math display="inline">\Lambda(A)</math>|If {{mvar|B}} is a subspace (or a convex set, or a balanced set) the same is true of <math display="inline">\Lambda^{-1}(B)</math>|In particular, the set: <math display="block">\Lambda^{-1}(\{0\}) = \{\mathbf x \in X: \Lambda \mathbf x = 0\} = {N}(\Lambda)</math> is a subspace of {{mvar|X}}, called the ''null space'' of {{tmath|\Lambda}}|list-style-type=lower-alpha}}</ref> for example, it maps a [[Plane (geometry)|plane]] through the origin in <math>V</math> to either a plane through the origin in {{tmath|W}}, a [[Line (geometry)|line]] through the origin in {{tmath|W}}, or just the origin in {{tmath|W}}. Linear maps can often be represented as [[matrix (mathematics)|matrices]], and simple examples include [[Rotations and reflections in two dimensions|rotation and reflection linear transformations]].

== Definition and first consequences ==

Let <math>V</math> and <math>W</math> be vector spaces over the same [[Field (mathematics)|field]] {{tmath|K}}, such as the [[real number|real]] or [[complex numbers]]. A [[function (mathematics)|function]] <math>f: V \to W</math> is said to be a ''linear map'' if for any two vectors <math display="inline">\mathbf{u}, \mathbf{v} \in V</math> and any scalar <math>c \in K</math> the following two conditions are satisfied: * [[Additive map|Additivity]] / operation of addition <math display=block>f(\mathbf{u} + \mathbf{v}) = f(\mathbf{u}) + f(\mathbf{v})</math> * [[Homogeneous function|Homogeneity]] of degree 1 / operation of scalar multiplication <math display=block>f(c \mathbf{u}) = c f(\mathbf{u})</math>

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 sides of the above examples) or after (the left sides of the examples) the operations of addition and scalar multiplication.

By [[Addition#Associativity|the associativity of the addition operation]] denoted as +, for any vectors <math display="inline"> \mathbf{u}_1, \ldots, \mathbf{u}_n \in V</math> and scalars {{tmath|c_1, \ldots, c_n \in K}}, the following equality holds:<ref>{{harvnb|Rudin|1991|page=14}}. Suppose now that {{mvar|X}} and {{mvar|Y}} are vector spaces ''over the same scalar field''. A mapping <math display="inline">\Lambda: X \to Y</math> is said to be ''linear'' if <math display="inline"> \Lambda(\alpha \mathbf x + \beta \mathbf y) = \alpha \Lambda \mathbf x + \beta \Lambda \mathbf y</math> for all <math display="inline">\mathbf x, \mathbf y \in X</math> and all scalars <math display="inline">\alpha</math> and {{tmath|\beta}}. Note that one often writes {{tmath|\Lambda \mathbf x}}, rather than {{tmath| \Lambda(\mathbf x) }}, when <math display="inline"> \Lambda</math> is linear.</ref><ref>{{harvnb|Rudin|1976|page=206}}. A mapping {{mvar|A}} of a vector space {{mvar|X}} into a vector space {{mvar|Y}} is said to be a ''linear transformation'' if: <math display="inline">A\left(\mathbf{x}_1 + \mathbf{x}_2\right) = A\mathbf{x}_1 + A\mathbf{x}_2,\ A(c\mathbf{x}) = c A\mathbf{x}</math> for all <math display="inline">\mathbf{x}, \mathbf{x}_1, \mathbf{x}_2 \in X</math> and all scalars {{mvar|c}}. Note that one often writes <math display="inline">A\mathbf{x}</math> instead of <math display="inline">A(\mathbf {x})</math> if {{mvar|A}} is linear.</ref> <math display="block">f(c_1 \mathbf{u}_1 + \cdots + c_n \mathbf{u}_n) = c_1 f(\mathbf{u}_1) + \cdots + c_n f(\mathbf{u}_n).</math> Thus a linear map is one which preserves [[linear combination]]s.

Denoting the zero elements of the vector spaces <math>V</math> and <math>W</math> by <math display="inline">\mathbf{0}_V</math> and <math display="inline">\mathbf{0}_W</math> respectively, it follows that {{tmath|1=f(\mathbf{0}_V) = \mathbf{0}_W}}. Let <math>c = 0</math> and <math display="inline">\mathbf{v} \in V</math> in the equation for homogeneity of degree 1: <math display="block">f(\mathbf{0}_V) = f(0\mathbf{v}) = 0f(\mathbf{v}) = \mathbf{0}_W.</math>

A linear map <math>V \to K</math> with <math>K</math> viewed as a one-dimensional vector space over itself is called a [[linear functional]].<ref>{{harvnb|Rudin|1991|page=14}}. Linear mappings of {{mvar|X}} onto its scalar field are called ''linear functionals''.</ref>

These statements generalize to any left-module <math display="inline">{}_R M</math> over a ring <math>R</math> without modification, and to any right-module upon reversing of the scalar multiplication.

== Examples == * The unique map of the form <math>T:\{\vec{0}\} \to \{\vec{0}\}</math> is linear. * A prototypical example that gives linear maps their name is a function {{tmath| f: \mathbb{R} \to \mathbb{R}: x \mapsto cx }}, of which the [[graph of a function|graph]] is a line through the origin.<ref>{{Cite web|title=terminology - What does 'linear' mean in Linear Algebra?|url=https://math.stackexchange.com/questions/62789/what-does-linear-mean-in-linear-algebra|access-date=2021-02-17|website=Mathematics Stack Exchange}}</ref>[[File:Linear_transformations_in_computer_graphics.svg|thumb|Examples of linear transformations used in computer graphics]] * More generally, any [[homothety]] <math display="inline">\mathbf{v} \mapsto c\mathbf{v}</math> centered in the origin of a vector space is a linear map (here {{mvar|c}} is a scalar). * The zero map <math display="inline">\mathbf x \mapsto \mathbf 0</math> between two vector spaces (over the same [[field (mathematics)|field]]) is linear. * The [[identity map]] on any module is a linear operator. * For real numbers, the map <math display="inline">x \mapsto x^2</math> is not linear. * For real numbers, the map <math display="inline">x \mapsto x + 1</math> is not linear (but is an [[affine transformation]]). * If <math>A</math> is a <math>m \times n</math> [[real matrix]], then <math>A</math> defines a linear map from <math>\R^n</math> to <math>\R^m</math> by sending a [[column vector]] <math>\mathbf x \in \R^n</math> to the column vector {{tmath|A \mathbf x \in \R^m}}. Conversely, any linear map between [[finite-dimensional]] vector spaces can be represented in this manner; see ''{{slink|#Matrices}}'', below. * If <math display="inline">f: V \to W</math> is an [[isometry]] between real [[normed space]]s such that <math display="inline"> f(0) = 0</math> then <math>f</math> is a linear map. This result is not necessarily true for complex normed space.{{sfn | Wilansky | 2013 | pp=21–26}} * [[Derivative|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]]s (a linear operator is a [[linear endomorphism]], that is, a linear map with the same [[Domain of a function|domain]] and [[codomain]]). Indeed, <math display="block">\frac{d}{dx} \left( a f(x) + b g(x) \right) = a \frac{d f(x)}{dx} + b \frac{d g( x)}{dx}.</math> * A definite [[integral]] over some [[interval (mathematics)|interval]] {{mvar|I}} is a linear map from the space of all real-valued integrable functions on {{mvar|I}} to {{tmath|\R}}. Indeed, <math display="block">\int_u^v \left(af(x) + bg(x)\right) dx = a\int_u^v f(x) dx + b\int_u^v g(x) dx . </math> * An indefinite [[integral]] (or [[antiderivative]]) with a fixed integration starting point defines a linear map from the space of all real-valued integrable functions on <math>\R</math> to the space of all real-valued, differentiable functions on {{tmath|\R}}. Without a fixed starting point, the antiderivative maps to the [[quotient space (linear algebra)|quotient space]] of the differentiable functions by the linear space of constant functions. * If <math>V</math> and <math>W</math> are finite-dimensional vector spaces over a field {{mvar|F}}, of respective dimensions {{mvar|m}} and {{mvar|n}}, then the function that maps linear maps <math display="inline">f: V \to W</math> to {{math|''n'' × ''m''}} matrices in the way described in ''{{slink|#Matrices}}'' (below) is a linear map, and even a [[linear isomorphism]]. * The [[expected value]] of a [[Random variable#Definition|random variable]] is a linear function of the random variable: for random variables <math>X</math> and <math>Y</math> we have <math>E[X + Y] = E[X] + E[Y]</math> and {{tmath|1= E[aX] = aE[X] }}. The [[conditional expectation]] is as well. But the [[variance]] of a random variable is not linear, because for instance {{tmath|1= \text{Var}(aX)=a^2\text{Var}(X) }}.

<gallery widths="180" heights="120"> File:Streckung eines Vektors.gif|The function <math display="inline">f:\R^2 \to \R^2</math> with <math display="inline">f(x, y) = (2x, y)</math> is a linear map. This function scales the <math display="inline">x</math> component of a vector by the factor {{tmath|2}}. File:Streckung der Summe zweier Vektoren.gif|The function <math display="inline">f(x, y) = (2x, y)</math> is additive: It does not matter whether vectors are first added and then mapped or whether they are mapped and finally added: <math display="inline">f(\mathbf a + \mathbf b) = f(\mathbf a) + f(\mathbf b)</math> File:Streckung homogenitaet Version 3.gif|The function <math display="inline">f(x, y) = (2x, y)</math> is homogeneous: It does not matter whether a vector is first scaled and then mapped or first mapped and then scaled: <math display="inline">f(\lambda \mathbf a) = \lambda f(\mathbf a)</math> </gallery>

=== Linear endomorphisms and isomorphisms === If a linear map is a [[bijection]] then it is called a '''{{visible anchor|Linear isomorphism|text=linear isomorphism}}'''. In the case where {{tmath|1= V = W }}, a linear map is called a '''linear endomorphism'''. Sometimes the term '''{{visible anchor|Linear operator|text=linear operator}}''' refers to this case,<ref>"Linear transformations of {{mvar|V}} into {{mvar|V}} are often called ''linear operators'' on {{mvar|V}}." {{harvnb|Rudin|1976|page=207}}</ref> but the term "linear operator" can have different meanings for different conventions.

=== Linear extensions ===

Often, a linear map is constructed by defining it on a subset of a vector space and then {{em|{{visible anchor|extending by linearity|extend by linearity}}}} to the [[linear span]] of the domain. Suppose <math>X</math> and <math>Y</math> are vector spaces and <math>f : S \to Y</math> is a [[Function (mathematics)|function]] defined on some subset {{tmath| S \subseteq X }}. Then a ''{{visible anchor|linear extension|Linear extension}} of <math>f</math> to <math>X,</math>'' if it exists, is a linear map <math>F : X \to Y</math> defined on <math>X</math> that [[Extension of a function|extends]] <math>f</math><ref group=note>One map <math>F</math> is said to [[Extension of a function|{{em|extend}}]] another map <math>f</math> if when <math>f</math> is defined at a point {{tmath| s }}, then so is <math>F</math> and <math>F(s) = f(s).</math></ref> (meaning that <math>F(s) = f(s)</math> for all {{tmath| s \in S }}) and takes its values from the codomain of {{tmath| f }}.{{sfn|Kubrusly|2001|p=57}} When the subset <math>S</math> is a vector subspace of <math>X</math> then a ({{tmath|Y}}-valued) linear extension of <math>f</math> to all of <math>X</math> is guaranteed to exist if (and only if) <math>f : S \to Y</math> is a linear map.{{sfn|Kubrusly|2001|p=57}} In particular, if <math>f</math> has a linear extension to <math>\operatorname{span} S,</math> then it has a linear extension to all of {{tmath| X }}.

The map <math>f : S \to Y</math> can be extended to a linear map <math>F : \operatorname{span} S \to Y</math> if and only if whenever <math>n > 0</math> is an integer, <math>c_1, \ldots, c_n</math> are scalars, and <math>s_1, \ldots, s_n \in S</math> are vectors such that {{tmath|1= 0 = c_1 s_1 + \cdots + c_n s_n }}, then necessarily {{tmath|1= 0 = c_1 f\left(s_1\right) + \cdots + c_n f\left(s_n\right) }}.{{sfn|Schechter|1996|pp=277–280}} If a linear extension of <math>f : S \to Y</math> exists then the linear extension <math>F : \operatorname{span} S \to Y</math> is unique and <math display=block>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)</math> holds for all {{tmath| n, c_1, \ldots, c_n }}, and <math>s_1, \ldots, s_n</math> as above.{{sfn|Schechter|1996|pp=277–280}} If <math>S</math> is linearly independent then every function <math>f : S \to Y</math> into any vector space has a linear extension to a (linear) map <math>\operatorname{span} S \to Y</math> (the converse is also true).

For example, if <math>X = \R^2</math> and <math>Y = \R</math> then the assignment <math>(1, 0) \to -1</math> and <math>(0, 1) \to 2</math> can be linearly extended from the linearly independent set of vectors <math>S := \{(1,0), (0, 1)\}</math> to a linear map on {{tmath|1= \operatorname{span}\{(1,0), (0, 1)\} = \R^2 }}. The unique linear extension <math>F : \R^2 \to \R</math> is the map that sends <math>(x, y) = x (1, 0) + y (0, 1) \in \R^2</math> to <math display=block>F(x, y) = x (-1) + y (2) = - x + 2 y.</math>

Every (scalar-valued) [[linear functional]] <math>f</math> defined on a [[vector subspace]] of a real or complex vector space <math>X</math> has a linear extension to all of {{tmath| X }}. Indeed, the [[Hahn–Banach theorem|Hahn–Banach dominated extension theorem]] even guarantees that when this linear functional <math>f</math> is dominated by some given [[seminorm]] <math>p : X \to \R</math> (meaning that <math>|f(m)| \leq p(m)</math> holds for all <math>m</math> in the domain of {{tmath| f }}) then there exists a linear extension to <math>X</math> that is also dominated by {{tmath| p }}.

== Matrices == {{main|Transformation matrix}}

If <math>V</math> and <math>W</math> are [[finite-dimensional]] vector spaces and a [[basis of a vector space|basis]] is defined for each vector space, then every linear map from <math>V</math> to <math>W</math> can be represented by a [[matrix (mathematics)|matrix]].<ref>{{harvnb|Rudin|1976|page=210}}</ref> This is useful because it allows concrete calculations. Matrices yield examples of linear maps: if <math>A</math> is a real <math>m \times n</math> matrix, then <math>f(\mathbf x) = A \mathbf x</math> describes a linear map <math>\R^n \to \R^m</math> (see [[Euclidean space]]).

Let <math>\{ \mathbf {v}_1, \ldots , \mathbf {v}_n \}</math> be a basis for {{tmath|V}}. Then every vector <math>\mathbf {v} \in V</math> is uniquely determined by the coefficients <math>c_1, \ldots , c_n</math> in the field {{tmath| \R }}: <math display="block">\mathbf{v} = c_1 \mathbf{v}_1 + \cdots + c_n \mathbf {v}_n.</math>

If <math display="inline">f: V \to W</math> is a linear map, <math display="block">f(\mathbf{v}) = f(c_1 \mathbf{v}_1 + \cdots + c_n \mathbf{v}_n) = c_1 f(\mathbf{v}_1) + \cdots + c_n f\left(\mathbf{v}_n\right),</math>

which implies that the function ''f'' is entirely determined by the vectors {{tmath| f(\mathbf {v}_1), \ldots , f(\mathbf {v}_n) }}. Now let <math>\{ \mathbf {w}_1, \ldots , \mathbf {w}_m \}</math> be a basis for {{tmath|W}}. Then we can represent each vector <math>f(\mathbf {v}_j)</math> as <math display="block">f\left(\mathbf{v}_j\right) = a_{1j} \mathbf{w}_1 + \cdots + a_{mj} \mathbf{w}_m.</math>

Thus, the function <math>f</math> is entirely determined by the values of {{tmath| a_{ij} }}. If we put these values into an <math>m \times n</math> matrix {{tmath|M}}, then we can conveniently use it to compute the vector output of <math>f</math> for any vector in {{tmath|V}}. To get {{tmath|M}}, every column <math>j</math> of <math>M</math> is a vector <math display="block">\begin{pmatrix} a_{1j} \\ \vdots \\ a_{mj} \end{pmatrix}</math> corresponding to <math>f(\mathbf {v}_j)</math> as defined above. To define it more clearly, for some column <math>j</math> that corresponds to the mapping {{tmath| f(\mathbf {v}_j) }}, <math display="block">\mathbf{M} = \begin{pmatrix} \ \cdots & a_{1j} & \cdots\ \\ & \vdots & \\ & a_{mj} & \end{pmatrix}</math> where <math>M</math> is the matrix of {{tmath| f }}. In other words, every column <math>j = 1, \ldots, n</math> has a corresponding vector <math>f(\mathbf {v}_j)</math> whose coordinates <math>a_{1j}, \cdots, a_{mj}</math> are the elements of column {{tmath| 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 <math display="inline">T</math> relative to {{tmath| B }}: <math display="inline">A</math> # Matrix for <math display="inline">T</math> relative to {{tmath| B' }}: <math display="inline">A'</math> # Transition matrix from <math display="inline">B'</math> to {{tmath| B }}: <math display="inline">P</math> # Transition matrix from <math display="inline">B</math> to {{tmath| B' }}: <math display="inline">P^{-1}</math>

[[File:Linear_transformation_visualization.svg|frame|The relationship between matrices in a linear transformation|none]]

Such that starting in the bottom left corner <math display="inline">\left[\mathbf{v}\right]_{B'}</math> and looking for the bottom right corner {{tmath|\textstyle \left[T\left(\mathbf{v}\right)\right]_{B'} }}, one would left-multiply—that is, {{tmath|1=\textstyle A'\left[\mathbf{v}\right]_{B'} = \left[T\left(\mathbf{v}\right)\right]_{B'} }}. The equivalent method would be the "longer" method going clockwise from the same point such that <math display="inline">\left[\mathbf{v}\right]_{B'}</math> is left-multiplied with {{tmath|P^{-1}AP}}, or {{tmath|1=\textstyle P^{-1}AP\left[\mathbf{v}\right]_{B'} = \left[T\left(\mathbf{v}\right)\right]_{B'} }}.

=== Examples in two dimensions === In two-[[dimension]]al space '''R'''2 linear maps are described by 2 × 2 [[matrix (mathematics)|matrices]]. These are some examples: * [[Rotation (mathematics)|rotation]] ** by 90 degrees counterclockwise: <math display="block">\mathbf{A} = \begin{pmatrix} 0 & -1\\ 1 & 0\end{pmatrix}</math> ** by an angle ''θ'' counterclockwise: <math display="block">\mathbf{A} = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}</math> * [[Reflection (mathematics)|reflection]] ** through the ''x'' axis: <math display="block">\mathbf{A} = \begin{pmatrix} 1 & 0\\ 0 & -1\end{pmatrix}</math> ** through the ''y'' axis: <math display="block">\mathbf{A} = \begin{pmatrix}-1 & 0\\ 0 & 1\end{pmatrix}</math> ** through a line making an angle ''θ'' with the origin: <math display="block">\mathbf{A} = \begin{pmatrix}\cos2\theta & \sin2\theta \\ \sin2\theta & -\cos2\theta \end{pmatrix}</math> * [[Scaling (geometry)|scaling]] by 2 in all directions: <math display="block">\mathbf{A} = \begin{pmatrix} 2 & 0\\ 0 & 2\end{pmatrix} = 2\mathbf{I}</math> * [[shear mapping|horizontal shear mapping]]: <math display="block">\mathbf{A} = \begin{pmatrix} 1 & m\\ 0 & 1\end{pmatrix}</math> * skew of the ''y'' axis by an angle ''θ'': <math display="block">\mathbf{A} = \begin{pmatrix} 1 & -\sin\theta\\ 0 & \cos\theta\end{pmatrix}</math> * [[squeeze mapping]]: <math display="block">\mathbf{A} = \begin{pmatrix} k & 0\\ 0 & \frac{1}{k}\end{pmatrix}</math> * [[Projection (linear algebra)|projection]] onto the ''y'' axis: <math display="block">\mathbf{A} = \begin{pmatrix} 0 & 0\\ 0 & 1\end{pmatrix}.</math>

If a linear map is only composed of rotation, reflection, and/or uniform scaling, then the linear map is a [[conformal linear transformation]].

== Vector space of linear maps ==

The composition of linear maps is linear: if <math>f: V \to W</math> and <math display="inline">g: W \to Z</math> are linear, then so is their [[Relation composition|composition]] {{tmath| g \circ f: V \to Z }}. It follows from this that the [[class (set theory)|class]] of all vector spaces over a given field ''K'', together with ''K''-linear maps as [[morphism]]s, forms a [[category (mathematics)|category]].

The [[inverse function|inverse]] of a linear map, when defined, is again a linear map.

If <math display="inline">f_1: V \to W</math> and <math display="inline">f_2: V \to W</math> are linear, then so is their [[pointwise]] sum {{tmath| f_1 + f_2 }}, which is defined by {{tmath|1= (f_1 + f_2)(\mathbf x) = f_1(\mathbf x) + f_2(\mathbf x) }}.

If <math display="inline">f: V \to W</math> is linear and <math display="inline">\alpha</math> is an element of the ground field {{tmath|K}}, then the map {{tmath| \alpha f }}, defined by {{tmath|1= (\alpha f)(\mathbf x) = \alpha (f(\mathbf x)) }}, is also linear.

Thus the set <math display="inline">\mathcal{L}(V, W)</math> of linear maps from <math display="inline">V</math> to <math display="inline">W</math> itself forms a vector space over {{tmath|K}},<ref>{{harvard citation text |Axler|2015}} p. 52, § 3.3</ref> sometimes denoted {{tmath| \operatorname{Hom}(V, W) }}.<ref>{{harvnb|Tu|2011|p=19|loc=§ 3.1}}</ref> Furthermore, in the case that {{tmath|1= V = W }}, this vector space, denoted {{tmath| \operatorname{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 finite-dimensional case, if bases have been chosen, then the composition of linear maps corresponds to the [[matrix multiplication]], the addition of linear maps corresponds to the [[matrix addition]], and the multiplication of linear maps with scalars corresponds to the multiplication of matrices with scalars.

=== Endomorphisms and automorphisms === {{main|Endomorphism|Automorphism}} A linear transformation <math display="inline">f : V \to V</math> is an [[endomorphism]] of <math display="inline">V</math>; the set of all such endomorphisms <math display="inline">\operatorname{End}(V)</math> together with addition, composition and scalar multiplication as defined above forms an [[associative algebra]] with identity element over the field <math display="inline">K</math> (and in particular a [[ring (algebra)|ring]]). The multiplicative identity element of this algebra is the [[identity map]] {{tmath| \operatorname{id}: V \to V }}.

An endomorphism of <math display="inline">V</math> that is also an [[isomorphism]] is called an [[automorphism]] of {{tmath|V}}. The composition of two automorphisms is again an automorphism, and the set of all automorphisms of <math display="inline">V</math> forms a [[group (math)|group]], the [[automorphism group]] of <math display="inline">V</math> which is denoted by <math display="inline">\operatorname{Aut}(V)</math> or {{tmath| \operatorname{GL}(V) }}. Since the automorphisms are precisely those [[endomorphisms]] which possess inverses under composition, <math display="inline">\operatorname{Aut}(V)</math> is the group of [[Unit (ring theory)|units]] in the ring {{tmath| \operatorname{End}(V) }}.

If <math display="inline">V</math> has finite dimension {{tmath|n}}, then <math display="inline"> \operatorname{End}(V)</math> is [[isomorphic]] to the [[associative algebra]] of all <math display="inline">n \times n</math> matrices with entries in {{tmath|K}}. The automorphism group of <math display="inline">V</math> is [[group isomorphism|isomorphic]] to the [[general linear group]] <math display="inline">\operatorname{GL}(n, K)</math> of all <math display="inline">n \times n</math> invertible matrices with entries in {{tmath|K}}.

== Kernel, image and the rank–nullity theorem == {{main|Kernel (linear algebra)|Image (mathematics)|Rank of a matrix}} If <math display="inline">f: V \to W</math> is linear, we define the [[kernel (linear operator)|kernel]] and the [[image (mathematics)|image]] or [[Range of a function|range]] of <math display="inline">f</math> by <math display="block">\begin{align} \ker(f) &= \{\,\mathbf x \in V: f(\mathbf x) = \mathbf 0\,\} \\ \operatorname{im}(f) &= \{\,\mathbf w \in W: \mathbf w = f(\mathbf x), \mathbf x \in V\,\} \end{align}</math>

<math display="inline">\ker(f)</math> is a [[Linear subspace|subspace]] of <math display="inline">V</math> and <math display="inline">\operatorname{im}(f)</math> is a subspace of {{tmath|W}}. The following [[dimension]] formula is known as the [[rank–nullity theorem]]:<ref>{{harvnb|Horn|Johnson|2013|loc=0.2.3 Vector spaces associated with a matrix or linear transformation, p. 6}}</ref> <math display="block">\dim(\ker( f )) + \dim(\operatorname{im}( f )) = \dim( V ).</math>

The number <math display="inline">\dim(\operatorname{im}(f))</math> is also called the [[rank of a matrix|rank]] of <math display="inline">f</math> and written as {{tmath| \operatorname{rank}(f) }}, or sometimes, {{tmath| \rho(f) }};<ref name=":0">{{Harvard citation text|Katznelson|Katznelson|2008}} p. 52, § 2.5.1</ref><ref name=":1">{{Harvard citation text|Halmos|1974}} p. 90, § 50</ref> the number <math display="inline">\dim(\ker(f))</math> is called the [[Kernel (matrix)#Subspace properties|nullity]] of <math display="inline">f</math> and written as <math display="inline">\operatorname{null}(f)</math> or {{tmath|\nu(f)}}.<ref name=":0" /><ref name=":1" /> If <math display="inline">V</math> and <math display="inline">W</math> are finite-dimensional, bases have been chosen and <math display="inline">f</math> is represented by the matrix {{tmath|A}}, then the rank and nullity of <math display="inline">f</math> are equal to the rank and nullity of the matrix {{tmath|A}}, respectively.

== Cokernel == {{main|Cokernel}}

A subtler invariant of a linear transformation <math display="inline">f : V \to W</math> is the [[cokernel|''co''kernel]], which is defined as <math display="block">\operatorname{coker}(f) := W/f(V) = W/\operatorname{im}(f).</math>

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 (linear algebra)|''quotient'' space]] of the ''target.'' Formally, one has the [[exact sequence]] <math display="block">0 \to \ker(f) \to V \to W \to \operatorname{coker}(f) \to 0.</math>

These can be interpreted thus: given a linear equation {{math|1=''f''('''v''') = '''w'''}} to solve, * the kernel is the space of ''solutions'' to the ''homogeneous'' equation {{math|1=''f''('''v''') = 0}}, and its dimension is the number of [[degrees of freedom]] in the space of solutions, if it is not empty; * the co-kernel is the space of [[wikt:constraint|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 {{math| ''W'' / ''f''(''V'')}} is the dimension of the target space minus the dimension of the image.

As a simple example, consider the map {{math|''f'' : '''R'''2 → '''R'''2}}, given by {{math|1=''f''(''x'', ''y'') = (0, ''y'')}}. Then for an equation {{math|1=''f''(''x'', ''y'') = (''a'', ''b'')}} to have a solution, we must have {{math|1=''a'' = 0}} (one constraint), and in that case the solution space is {{math|(''x'', ''b'')}} or equivalently stated, {{math| (0, ''b'') + (''x'', 0)}}, (one degree of freedom). The kernel may be expressed as the subspace {{math|(''x'', 0) < ''V''}}: the value of {{math|''x''}} is the freedom in a solution – while the cokernel may be expressed via the map {{math|''W'' → '''R'''}}, {{tmath| (a, b) \mapsto (a) }}: given a vector {{math|(''a'', ''b'')}}, the value of {{math|''a''}} is the ''obstruction'' to there being a solution.

An example illustrating the infinite-dimensional case is afforded by the map {{math|''f'' : '''R'''∞ → '''R'''∞}}, <math display="inline">\left\{a_n\right\} \mapsto \left\{b_n\right\}</math> with {{math|1=''b''1 = 0}} and {{math|1=''b''''n'' + 1 = ''an''}} for {{math|''n'' > 0}}. Its image consists of all sequences with first element {{math|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 [[cardinal number#Cardinal addition|sum]] as the rank and the dimension of the co-kernel ({{tmath|1= \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]] have the same dimension ({{math|0 ≠ 1}}). The reverse situation obtains for the map {{math|''h'' : '''R'''∞ → '''R'''∞}}, <math display="inline">\left\{a_n\right\} \mapsto \left\{c_n\right\}</math> with {{math|1=''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: <math display="block">\operatorname{ind}(f) := \dim(\ker(f)) - \dim(\operatorname{coker}(f)),</math> 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]] of the 2-term complex {{math|0 → ''V'' → ''W'' → 0}}. In [[operator theory]], the index of [[Fredholm operator]]s is an object of study, with a major result being the [[Atiyah–Singer index theorem]].<ref>{{SpringerEOM|title=Index theory|id=Index_theory&oldid=23864|first=Victor|last=Nistor}}: "The main question in index theory is to provide index formulas for classes of Fredholm operators ... Index theory has become a subject on its own only after M. F. Atiyah and I. Singer published their index theorems"</ref>

== 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 {{mvar|V}} and {{mvar|W}} denote vector spaces over a field {{mvar|F}} and let {{math|''T'' : ''V'' → ''W''}} be a linear map.

=== Monomorphism === {{mvar|T}} is said to be ''[[injective]]'' or a ''[[monomorphism]]'' if any of the following equivalent conditions are true: # {{mvar|T}} is [[injective|one-to-one]] as a map of [[set (mathematics)|sets]]. # {{math|1=ker ''T'' = {0''V''} }} # {{math|1=dim(ker ''T'') = 0}} # {{mvar|T}} is [[monic morphism|monic]] or left-cancellable, which is to say, for any vector space {{mvar|U}} and any pair of linear maps {{math|''R'': ''U'' → ''V''}} and {{math|''S'' : ''U'' → ''V''}}, the equation {{math|1=''TR'' = ''TS''}} implies {{math|1=''R'' = ''S''}}. # {{mvar|T}} is [[inverse (ring theory)|left-invertible]], which is to say there exists a linear map {{math|''S'' : ''W'' → ''V''}} such that {{math|''ST''}} is the [[identity map]] on {{mvar|V}}.

=== Epimorphism === {{mvar|T}} is said to be ''[[surjective]]'' or an ''[[epimorphism]]'' if any of the following equivalent conditions are true: # {{mvar|T}} is [[onto]] as a map of sets. # {{math|1=[[cokernel|coker]] ''T'' = {0''W''} }} # {{mvar|T}} is [[epimorphism|epic]] or right-cancellable, which is to say, for any vector space {{mvar|U}} and any pair of linear maps {{math|''R'' : ''W'' → ''U''}} and {{math|''S'' : ''W'' → ''U''}}, the equation {{math|1=''RT'' = ''ST''}} implies {{math|1=''R'' = ''S''}}. # {{mvar|T}} is [[inverse (ring theory)|right-invertible]], which is to say there exists a linear map {{math|''S'' : ''W'' → ''V''}} such that {{math|''TS''}} is the [[identity map]] on {{mvar|W}}.

=== Isomorphism === {{mvar|T}} is said to be an ''[[isomorphism]]'' if it is both left- and right-invertible. This is equivalent to {{mvar|T}} being both one-to-one and onto (a [[bijection]] of sets) or also to {{mvar|T}} being both epic and monic, and so being a [[bimorphism]]. {{pb}} If {{math|''T'' : ''V'' → ''V''}} is an endomorphism, then: * If, for some positive integer {{mvar|n}}, the {{mvar|n}}th iterate of {{mvar|T}}, {{math|''T''''n''}}, is identically zero, then {{mvar|T}} is said to be [[nilpotent]]. * If {{math|1=''T''2 = ''T''}}, then {{mvar|T}} is said to be [[idempotent]] * If {{math|1=''T'' = ''kI''}}, where {{mvar|k}} is some scalar, then {{mvar|T}} is said to be a scaling transformation or scalar multiplication map; see [[scalar matrix]].

== Change of basis == {{main|Basis (linear algebra)|Change of basis}} Given a linear map which is an [[endomorphism]] whose matrix is {{math|''A''}}, in the basis {{math|''B''}} of the space it transforms vector coordinates {{math|[''u'']}} as {{math|1=[''v''] = ''A''[''u'']}}. As vectors change with the inverse of {{math|''B''}} (vectors coordinates are [[Covariance and contravariance of vectors|contravariant]]) its inverse transformation is {{math|1=[''v''] = ''B''[''v''′]}}.

Substituting this in the first expression <math display="block">B\left[v'\right] = AB\left[u'\right]</math> hence <math display="block">\left[v'\right] = B^{-1}AB\left[u'\right] = A'\left[u'\right].</math>

Therefore, the matrix in the new basis is {{math|1=''A''′ = ''B''−1''AB''}}, being {{math|''B''}} the matrix of the given basis.

Therefore, linear maps are said to be 1-co- 1-contra-[[covariance and contravariance of vectors|variant]] objects, or type (1, 1) [[tensor]]s.

== Continuity == {{main|Continuous linear operator|Discontinuous linear map}}

A ''linear transformation'' between [[topological vector space]]s, for example [[normed space]]s, may be [[continuous function (topology)|continuous]]. If its domain and codomain are the same, it will then be a [[continuous linear operator]]. A linear operator on a normed linear space is continuous if and only if it is [[bounded operator|bounded]], for example, when the domain is finite-dimensional.<ref>{{harvnb|Rudin|1991|page=15}}

'''1.18 Theorem''' ''Let <math display="inline">\Lambda</math> be a linear functional on a topological vector space {{mvar|X}}. Assume {{tmath| \Lambda \mathbf x \neq 0 }} for some {{tmath| \mathbf x \in X }}. Then each of the following four properties implies the other three'': {{ordered list |list-style-type=lower-alpha |<math display="inline">\Lambda</math> is continuous |The null space <math display="inline">N(\Lambda)</math> is closed. |<math display="inline">N(\Lambda)</math> is not dense in {{mvar|X}}. |<math display="inline">\Lambda</math> is bounded in some neighbourhood {{mvar|V}} of {{math|0}}.}}</ref> An infinite-dimensional domain may have [[discontinuous linear operator]]s.

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 {{math|0}} is {{math|0}}). For a specific example, {{math|sin(''nx'')/''n''}} converges to {{math|0}}, but its derivative {{math|cos(''nx'')}} does not, so differentiation is not continuous at {{math|0}} (and by a variation of this argument, it is not continuous anywhere).

== Applications == A specific application of linear maps is for [[geometric transformation]]s, such as those performed in [[computer graphics]], where the translation, rotation and scaling of 2D or 3D objects is performed by the use of a [[transformation matrix]]. 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.

== See also == {{wikibooks|Linear Algebra/Linear Transformations}}

* {{annotated link|Additive map}} * {{annotated link|Antilinear map}} * {{annotated link|Bent function}} * {{annotated link|Bounded operator}} * {{annotated link|Cauchy's functional equation}} * {{annotated link|Continuous linear operator}} * {{annotated link|Linear functional}} * {{annotated link|Linear isometry}} * [[Category of matrices]] * [[Quasilinearization]]

== Notes ==

== Bibliography ==

* {{cite book|last=Axler|first=Sheldon Jay|title=Linear Algebra Done Right|publisher=[[Springer Science+Business Media|Springer]]|year=2015|isbn=978-3-319-11079-0|edition=3rd|author-link=Sheldon Axler}} * {{cite book |last1=Bronshtein |first1= I. N. |last2= Semendyayev |first2= K. A. |year=2004 |title= [[Handbook of Mathematics]] |edition= 4th |location= New York |publisher= Springer-Verlag |isbn=3-540-43491-7}} * {{cite book|last=Halmos|first=Paul Richard|title=Finite-Dimensional Vector Spaces|publisher=[[Springer Science+Business Media|Springer]]|year=1974|isbn=0-387-90093-4|edition=2nd|author-link=Paul Halmos|orig-year=1958}} * {{cite book | last1=Horn | first1=Roger A. | last2=Johnson | first2=Charles R. | title=Matrix Analysis |edition=Second |publisher=[[Cambridge University Press]] | isbn=978-0-521-83940-2 | year=2013 }} * {{cite book|last1=Katznelson|first1=Yitzhak|title=A (Terse) Introduction to Linear Algebra|last2=Katznelson|first2=Yonatan R.|publisher=[[American Mathematical Society]]|year=2008|isbn=978-0-8218-4419-9|author-link1=Yitzhak Katznelson}} * {{cite book|last=Kubrusly|first=Carlos|title=Elements of operator theory|publisher=Birkhäuser|publication-place=Boston|year=2001|isbn=978-1-4757-3328-0|oclc=754555941}}  * {{citation | last1=Lang | first1=Serge | author1-link=Serge Lang | title=Linear Algebra | publisher=[[Springer-Verlag]] | location=New York | isbn=0-387-96412-6 |edition=Third | year=1987}} * {{Rudin Walter Functional Analysis|edition=1}}  * {{cite book|last=Rudin|first=Walter|author-link=Walter Rudin|year=1976|edition=3rd|series=Walter Rudin Student Series in Advanced Mathematics|location=New York|url=https://archive.org/details/PrinciplesOfMathematicalAnalysis|title=Principles of Mathematical Analysis|publisher=McGraw–Hill|isbn=978-0-07-054235-8}} * {{Rudin Walter Functional Analysis|edition=2}}  * {{Schaefer Wolff Topological Vector Spaces|edition=2}}  * {{Schechter Handbook of Analysis and Its Foundations}}  * {{Swartz An Introduction to Functional Analysis}}  * {{cite book|last=Tu|first=Loring W.|title=An Introduction to Manifolds|title-link=An Introduction to Manifolds|publisher=[[Springer Science+Business Media|Springer]]|year=2011|isbn=978-0-8218-4419-9|edition=2nd|author-link=Loring W. Tu}} * {{Wilansky Modern Methods in Topological Vector Spaces|edition=1}}

[[Category:Abstract algebra]] [[Category:Functions and mappings]] [[Category:Linear operators| ]] [[Category:Transformation (function)]]