Editing Linear map (section)

==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]] <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]]).
* 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]] 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 <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]] (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 <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]] 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]] 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]] 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 [[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 <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]] <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>