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{{Short description|Mathematical function, in linear algebra}}
In [[Mathematics|mathematics]], and more specifically in [[Linear algebra|linear algebra]], a '''linear map''' (also called a '''linear mapping''', '''linear transformation''', '''vector space homomorphism''', or in some contexts '''linear function''') is a [[Map (mathematics)|mapping]] <math>V \to W</math> between two  [[Vector space|vector space]]s that preserves the operations of [[Vector addition|vector addition]] and [[Scalar multiplication|scalar multiplication]]. The same names and the same definition are also used for the more general case of [[Module (mathematics)|modules]] over a [[Ring (mathematics)|ring]]; see [[Module homomorphism]].

If a linear map is a [[Bijection|bijection]] then it is called a '''{{visible anchor|Linear isomorphism|text=linear isomorphism}}'''. In the case where <math>V = W</math>, 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: for example, it can be used to emphasize that <math>V</math> and <math>W</math> are [[Real number|real]] vector spaces (not necessarily with <math>V = W</math>),{{citation needed|date=November 2020}} or it can be used to emphasize that <math>V</math> is a [[Function space|function space]], which is a common convention in [[Functional analysis|functional analysis]].<ref>Let {{mvar|V}} and {{mvar|W}} be two real vector spaces. A mapping a from {{mvar|V}} into {{mvar|W}} Is called a 'linear mapping' or 'linear transformation' or 'linear operator' [...] from {{mvar|V}} into {{mvar|W}}, if <br /> <math display="inline">a(\mathbf u + \mathbf v) = a \mathbf u + a \mathbf v</math> for all <math display="inline">\mathbf u,\mathbf v \in V</math>, <br /> <math display="inline"> a(\lambda \mathbf u) = \lambda a \mathbf u </math> for all <math>\mathbf u \in V</math> and all real {{mvar|λ}}. {{harvnb|Bronshtein|Semendyayev|2004|page=316}}</ref> Sometimes the term ''[[Linear function|linear function]]'' has the same meaning as ''linear map'', while in [[Mathematical analysis|analysis]] it does not.

A linear map from <math>V</math>  to <math>W</math> always maps the origin of <math>V</math>  to the origin of <math>W</math>. Moreover, it maps  [[Linear subspace|linear subspace]]s in <math>V</math> onto linear subspaces in <math>W</math> (possibly of a lower [[Dimension (vector space)|dimension]]);<ref>{{harvnb|Rudin|1991|page=14}}<br />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|convex set]], or a [[Balanced set|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 <math display="inline">\Lambda</math>.|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 <math>W</math>, a [[Line (geometry)|line]] through the origin in <math>W</math>, or just the origin in <math>W</math>. 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]].

In the language of [[Category theory|category theory]], linear maps are the [[Morphism|morphism]]s of vector spaces.

==Definition and first consequences==

Let <math>V</math> and <math>W</math> be vector spaces over the same [[Field (mathematics)|field]] <math>K</math>. 
A 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 hand sides of the above examples) or after (the left hand 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 <math display="inline">c_1, \ldots, c_n \in K,</math> 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 <math display="inline">\beta</math>. Note that one often writes <math display="inline">\Lambda \mathbf x</math>, rather than <math display="inline">\Lambda(\mathbf x)</math>, 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|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 <math display="inline">f(\mathbf{0}_V) = \mathbf{0}_W.</math> 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==

* A prototypical example that gives linear maps their name is a function <math>f: \mathbb{R} \to \mathbb{R}: x \mapsto cx</math>, 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>
* More generally, any [[Homothety|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 function|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|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 <math>A \mathbf x \in \R^m</math>. Conversely, any linear map between finite-dimensional vector spaces can be represented in this manner; see the {{slink||Matrices}}, below.
* If <math display="inline">f: V \to W</math> is an [[Isometry|isometry]] between real normed spaces 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 functions (a linear operator is a  linear endomorphism, that is, a linear map with the same [[Domain of a function|domain]] and [[Codomain|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|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 <math>\R</math>. 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|integral]] (or [[Antiderivative|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 <math>\R</math>. 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|expected value]] of a [[Random variable#Definition|random variable]] (which is in fact a function, and as such an element of a vector space) is linear, as for random variables <math>X</math> and <math>Y</math> we have <math>E[X + Y] = E[X] + E[Y]</math> and <math>E[aX] = aE[X]</math>, but the [[Variance|variance]] of a random variable is not linear.

<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 <math display="inline">2</math>.
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 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|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 <math>S \subseteq X.</math> 
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 extends <math>f</math><ref group=note>One map <math>F</math> is said to {{em|extend}} another map <math>f</math> if when <math>f</math> is defined at a point <math>s,</math> then so is <math>F</math> and <math>F(s) = f(s).</math></ref> (meaning that <math>F(s) = f(s)</math> for all <math>s \in S</math>) and takes its values from the codomain of <math>f.</math>{{sfn|Kubrusly|2001|p=57}} 
When the subset <math>S</math> is a vector subspace of <math>X</math> then a (<math>Y</math>-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 <math>X.</math>

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 <math>0 = c_1 s_1 + \cdots + c_n s_n,</math> then necessarily <math>0 = c_1 f\left(s_1\right) + \cdots + c_n f\left(s_n\right).</math>{{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 <math>n, c_1, \ldots, c_n,</math> 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 <math>\operatorname{span}\{(1,0), (0, 1)\} = \R^2.</math> 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 <math>X.</math> 
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|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 <math>f</math>) then there exists a linear extension to <math>X</math> that is also dominated by <math>p.</math>

==Matrices==
{{Main|Transformation matrix}}

If <math>V</math> and <math>W</math> are finite-dimensional vector spaces and a 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}}

Suppose <math display="inline">\left\{\mathbf{x}_1, \ldots, \mathbf{x}_n\right\}</math> and <math display="inline">\left\{\mathbf{y}_1, \ldots, \mathbf{y}_m\right\}</math> are bases of vector spaces {{mvar|X}} and {{mvar|Y}}, respectively. Then every <math display="inline">A \in L(X, Y)</math> determines a set of numbers <math display="inline">a_{i,j}</math> such that
<math display="block">A\mathbf{x}_j = \sum_{i=1}^m a_{i,j}\mathbf{y}_i\quad (1 \leq j \leq n).</math>

It is convenient to represent these numbers in a rectangular array of {{mvar|m}} rows and {{mvar|n}} columns, called an {{mvar|m}} ''by'' {{mvar|n}} ''matrix'':
<math display="block">[A] = \begin{bmatrix}
  a_{1,1} & a_{1,2} & \ldots & a_{1,n} \\
  a_{2,1} & a_{2,2} & \ldots & a_{2,n} \\ 
   \vdots &  \vdots & \ddots &  \vdots \\
  a_{m,1} & a_{m,2} & \ldots & a_{m,n}
\end{bmatrix}</math>

Observe that the coordinates <math display="inline">a_{i,j}</math> of the vector <math display="inline">A\mathbf{x}_j</math> (with respect to the basis <math display="inline">\{\mathbf{y}_1, \ldots, \mathbf{y}_m\}</math>) appear in the ''j''<sup>th</sup> column of <math display="inline">[A]</math>. The vectors <math display="inline">A\mathbf{x}_j</math> are therefore sometimes called the ''column vectors'' of <math display="inline">[A]</math>. With this terminology, the ''range'' of {{mvar|A}} ''is spanned by the column vectors of <math display="inline">[A]</math>''.
</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 <math>V</math>. 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 <math>\R</math>:
<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 <math>f(\mathbf {v}_1), \ldots , f(\mathbf {v}_n)</math>. Now let <math>\{ \mathbf {w}_1, \ldots , \mathbf {w}_m \}</math> be a basis for <math>W</math>.  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 <math>a_{ij}</math>. If we put these values into an <math>m \times n</math> matrix <math>M</math>, then we can conveniently use it to compute the vector output of <math>f</math> for any vector in <math>V</math>.  To get <math>M</math>, 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 <math>f(\mathbf {v}_j)</math>,
<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 <math>f</math>. 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 <math>j</math>. 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 <math display="inline">B</math>: <math display="inline">A</math>
# Matrix for <math display="inline">T</math> relative to <math display="inline">B'</math>: <math display="inline">A'</math>
# Transition matrix from <math display="inline">B'</math> to <math display="inline">B</math>: <math display="inline">P</math>
# Transition matrix from <math display="inline">B</math> to <math display="inline">B'</math>: <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 <math display="inline">\left[T\left(\mathbf{v}\right)\right]_{B'}</math>, one would left-multiply—that is, <math display="inline">A'\left[\mathbf{v}\right]_{B'} = \left[T\left(\mathbf{v}\right)\right]_{B'}</math>. 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 <math display="inline">P^{-1}AP</math>, or <math display="inline">P^{-1}AP\left[\mathbf{v}\right]_{B'} = \left[T\left(\mathbf{v}\right)\right]_{B'}</math>.

===Examples in two dimensions===
In two-[[Physics:Dimension|dimension]]al space '''R'''<sup>2</sup> 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|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|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 composition <math display="inline">g \circ f: V \to Z</math>. 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|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|pointwise]] sum <math>f_1 + f_2</math>, which is defined by <math>(f_1 + f_2)(\mathbf x) = f_1(\mathbf x) + f_2(\mathbf x)</math>.

If <math display="inline">f: V \to W</math> is linear and <math display="inline">\alpha</math> is an element of the ground field <math display="inline">K</math>, then the map <math display="inline">\alpha f</math>, defined by <math display="inline">(\alpha f)(\mathbf x) = \alpha (f(\mathbf x))</math>, 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 <math display="inline">K</math>,<ref>{{Harvard citation text |Axler|2015}} p. 52, § 3.3</ref> sometimes denoted <math display="inline">\operatorname{Hom}(V, W)</math>.<ref>{{Harvard citation text|Tu|2011}}, p. 19, § 3.1</ref> Furthermore, in the case that <math display="inline">V = W</math>, this vector space, denoted <math display="inline"> \operatorname{End}(V)</math>, is an [[Associative algebra|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|matrix multiplication]], the addition of linear maps corresponds to the [[Matrix addition|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|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|associative algebra]] with identity element over the field <math display="inline">K</math> (and in particular a ring). The multiplicative identity element of this algebra is the [[Identity function|identity map]] <math display="inline">\operatorname{id}: V \to V</math>.

An endomorphism of <math display="inline">V</math> that is also an [[Isomorphism|isomorphism]] is called an [[Automorphism|automorphism]] of <math display="inline">V</math>. The composition of two automorphisms is again an automorphism, and the set of all automorphisms of <math display="inline">V</math> forms a group, the automorphism group of <math display="inline">V</math> which is denoted by <math display="inline">\operatorname{Aut}(V)</math> or <math display="inline">\operatorname{GL}(V)</math>. 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 <math display="inline">\operatorname{End}(V)</math>.

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

==Kernel, image and the rank–nullity theorem==
If <math display="inline">f: V \to W</math> is linear, we define the 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 <math display="inline">W</math>.  The following [[Physics:Dimension|dimension]] formula is known as the [[Rank–nullity theorem|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 <math display="inline">f</math> and written as <math display="inline">\operatorname{rank}(f)</math>, or sometimes, <math display="inline">\rho(f)</math>;<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 nullity of <math display="inline">f</math> and written as <math display="inline">\operatorname{null}(f)</math> or <math display="inline">\nu(f)</math>.<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 <math display="inline">A</math>, then the rank and nullity of <math display="inline">f</math> are equal to the rank and nullity of the matrix <math display="inline">A</math>, 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|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 ''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]] 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'': '''R'''<sup>2</sup> → '''R'''<sup>2</sup>, 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''', <math display="inline"> (a, b) \mapsto (a)</math>: 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'''<sup>∞</sup> → '''R'''<sup>∞</sup>, <math display="inline">\left\{a_n\right\} \mapsto \left\{b_n\right\}</math> with ''b''<sub>1</sub> = 0 and ''b''<sub>''n'' + 1</sub> = ''a<sub>n</sub>'' 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 [[Cardinal number#Cardinal addition|sum]] as the rank and the dimension of the co-kernel (<math display="inline">\aleph_0 + 0 = \aleph_0 + 1</math>), but in the infinite-dimensional case it cannot be inferred that the kernel and the co-kernel of an [[Endomorphism|endomorphism]] have the same dimension (0 ≠ 1). The reverse situation obtains for the map ''h'': '''R'''<sup>∞</sup> → '''R'''<sup>∞</sup>, <math display="inline">\left\{a_n\right\} \mapsto \left\{c_n\right\}</math> with ''c<sub>n</sub>'' = ''a''<sub>''n'' + 1</sub>. 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 0 → ''V'' → ''W'' → 0. In [[Operator theory|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|monomorphism]]'' if any of the following equivalent conditions are true:
# {{mvar|T}} is one-to-one as a map of [[Set (mathematics)|sets]].
# {{math|1=ker ''T'' = {0<sub>''V''</sub>} }}
# {{math|1=dim(ker ''T'') = 0}}
# {{mvar|T}} is 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 left-invertible, which is to say there exists a linear map {{math|''S'': ''W'' → ''V''}} such that {{math|''ST''}} is the [[Identity function|identity map]] on {{mvar|V}}.

===Epimorphism===
{{mvar|T}} is said to be ''surjective'' or an ''[[Epimorphism|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<sub>''W''</sub>} }}
# {{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 right-invertible, which is to say there exists a linear map {{math|''S'': ''W'' → ''V''}} such that {{math|''TS''}} is the [[Identity function|identity map]] on {{mvar|W}}.

===Isomorphism===
{{mvar|T}} is said to be an ''[[Isomorphism|isomorphism]]'' if it is both left- and right-invertible. This is equivalent to {{mvar|T}} being both one-to-one and onto (a [[Bijection|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''<sup>''n''</sup>}}, is identically zero, then {{mvar|T}} is said to be [[Nilpotent|nilpotent]].
* If {{math|1=''T''<sup>2</sup> = ''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|endomorphism]] whose matrix is ''A'', in the basis ''B'' of the space it transforms vector coordinates [u] as [v] = ''A''[u]. As vectors change with the inverse of ''B'' (vectors are [[Covariance and contravariance of vectors|contravariant]]) its inverse transformation is [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 ''A′'' = ''B''<sup>−1</sup>''AB'', being ''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|tensor]]s.

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

A ''linear transformation'' between [[Topological vector space|topological vector space]]s, for example normed spaces, may be continuous. If its domain and codomain are the same, it will then be a [[Continuous linear operator|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 <math display="inline">\Lambda \mathbf x \neq 0</math> for some <math display="inline">\mathbf x \in X</math>. 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 0.}}</ref> 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, {{math|sin(''nx'')/''n''}} converges to 0, but its derivative {{math|cos(''nx'')}} 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 transformation|geometric transformation]]s, such as those performed in [[Computer graphics|computer graphics]], where the translation, rotation and scaling of 2D or 3D objects is performed by the use of a [[Transformation matrix|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==

* {{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}}

==Notes==

{{reflist}}

{{reflist|group=note}}

==Bibliography==

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{{linear algebra}}
{{tensors}}

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

{{Sourceattribution|Linear map}}