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Linear map
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===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>
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