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Linear form
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{{Short description|Linear map from a vector space to its field of scalars}} In [[mathematics]], a '''linear form''' (also known as a '''linear functional''',<ref>{{Harvard citation text|Axler|2015}} p. 101, §3.92</ref> a '''one-form''', or a '''covector''') is a [[linear map]]<ref group=nb>In some texts the roles are reversed and vectors are defined as linear maps from covectors to scalars</ref> from a [[vector space]] to its [[field (mathematics)|field]] of [[scalar (mathematics)|scalars]] (often, the [[real number]]s or the [[complex number]]s). If {{mvar|V}} is a vector space over a field {{mvar|k}}, the set of all linear functionals from {{mvar|V}} to {{mvar|k}} is itself a vector space over {{mvar|k}} with addition and scalar multiplication defined [[pointwise]]. This space is called the [[dual space]] of {{mvar|V}}, or sometimes the '''algebraic dual space''', when a [[topological dual space]] is also considered. It is often denoted {{math|Hom(''V'', ''k'')}},<ref name=":0">{{Harvard citation text|Tu|2011}} p. 19, §3.1</ref> or, when the field {{mvar|k}} is understood, <math>V^*</math>;<ref>{{Harvard citation text|Katznelson|Katznelson|2008}} p. 37, §2.1.3</ref> other notations are also used, such as <math>V'</math>,<ref>{{Harvard citation text|Axler|2015}} p. 101, §3.94</ref><ref>{{Harvtxt|Halmos|1974}} p. 20, §13</ref> <math>V^{\#}</math> or <math>V^{\vee}.</math><ref name=":0" /> When vectors are represented by [[column vector]]s (as is common when a [[basis (linear algebra)|basis]] is fixed), then linear functionals are represented as [[row vector]]s, and their values on specific vectors are given by [[matrix product]]s (with the row vector on the left).
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