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| In [[mathematics]], the '''(formal) complex conjugate''' of a [[complex numbers|complex]] [[vector space]] <math>V\,</math> is the complex vector space <math>\overline V</math> consisting of all formal [[complex conjugate]]s of elements of <math>V\,</math>. That is, <math>\overline V</math> is a vector space whose elements are in [[bijection|one-to-one correspondence]] with the elements of <math>V\,</math>:
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| :<math>\overline V = \{\overline v \mid v \in V\},</math>
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| with the following rules for [[addition]] and [[scalar multiplication]]:
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| :<math>\overline v + \overline w = \overline{\,v+w\,}\quad\text{and}\quad\alpha\,\overline v = \overline{\,\overline \alpha \,v\,}.</math>
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| Here <math>v\,</math> and <math>w\,</math> are vectors in <math>V\,</math>, <math>\alpha\,</math> is a complex number, and <math>\overline\alpha</math> denotes the complex conjugate of <math>\alpha\,</math>.
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| More concretely, the complex conjugate vector space is the same underlying ''real'' vector space (same set of points, same vector addition and real scalar multiplication) with the conjugate [[linear complex structure]] ''J'' (different multiplication by ''i'').
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| ==Antilinear maps==
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| If <math>V\,</math> and <math>W\,</math> are complex vector spaces, a function <math>f\colon V \to W\,</math> is [[antilinear]] if
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| :<math>f(v+v') = f(v) + f(v')\quad\text{and}\quad f(\alpha v) = \overline\alpha \, f(v)</math>
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| for all <math>v,v'\in V\,</math> and <math>\alpha\in\mathbb{C}</math>.
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| One reason to consider the vector space <math>\overline V</math> is that it makes antilinear maps into [[linear map]]s. Specifically, if <math>f\colon V \to W\,</math> is an antilinear map, then the corresponding map <math>\overline V \to W</math> defined by
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| :<math>\overline v \mapsto f(v)</math>
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| is linear. Conversely, any linear map defined on <math>\overline V</math> gives rise to an antilinear map on <math>V\,</math>.
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| One way of thinking about this correspondence is that the map <math>C\colon V \to \overline V</math> defined by
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| :<math>C(v) = \overline v</math> | |
| is an antilinear bijection. Thus if <math>f\colon \overline V \to W</math> is linear, then [[Function composition|composition]] <math>f \circ C\colon V \to W\,</math> is antilinear, and ''vice versa''.
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| ==Conjugate linear maps==
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| Any linear map <math>f \colon V \to W\,</math> induces a '''conjugate linear map''' <math>\overline f \colon \overline V \to \overline W</math>, defined by the formula
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| :<math>\overline f (\overline v) = \overline{\,f(v)\,}.</math> | |
| The conjugate linear map <math>\overline f</math> is linear. Moreover, the [[identity function|identity map]] on <math>V\,</math> induces the identity map <math>\overline V</math>, and
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| :<math>\overline f \circ \overline g = \overline{\,f \circ g\,}</math> | |
| for any two linear maps <math>f\,</math> and <math>g\,</math>. Therefore, the rules <math>V\mapsto \overline V</math> and <math>f\mapsto\overline f</math> define a [[functor]] from the [[category theory|category]] of complex vector spaces to itself.
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| If <math>V\,</math> and <math>W\,</math> are finite-dimensional and the map <math>f\,</math> is described by the complex [[matrix (mathematics)|matrix]] <math>A\,</math> with respect to the [[basis of a vector space|bases]] <math>\mathcal B</math> of <math>V\,</math> and <math>\mathcal C</math> of <math>W\,</math>, then the map <math>\overline f</math> is described by the complex conjugate of <math>A\,</math> with respect to the bases <math>\overline{\mathcal B}</math> of <math>\overline V</math> and <math>\overline{\mathcal C}</math> of <math>\overline W</math>.
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| ==Structure of the conjugate==
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| The vector spaces <math>V\,</math> and <math>\overline V</math> have the same [[dimension of a vector space|dimension]] over the complex numbers and are therefore [[isomorphism|isomorphic]] as complex vector spaces. However, there is no [[natural isomorphism]] from <math>V\,</math> to <math>\overline V</math>. (The map <math>C\,</math> is not an isomorphism, since it is antilinear.)
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| The double conjugate <math>\overline{\overline V}</math> is naturally isomorphic to <math>V\,</math>, with the isomorphism <math>\overline{\overline V} \to V</math> defined by
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| :<math>\overline{\overline v} \mapsto v.</math>
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| Usually the double conjugate of <math>V\,</math> is simply identified with <math>V\,</math>.
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| == Complex conjugate of a Hilbert space ==
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| Given a [[Hilbert space]] <math>\mathcal{H}</math> (either finite or infinite dimensional), its complex conjugate <math>\overline{\mathcal{H}}</math> is the same vector space as its [[continuous dual space]] <math>\mathcal{H}'</math>.
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| There is one-to-one antilinear correspondence between continuous linear functionals and vectors.
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| In other words, any continuous [[linear functional]] on <math>\mathcal{H}</math> is an inner multiplication to some fixed vector, and vice versa.
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| Thus, the complex conjugate to a vector <math>v</math>, particularly in finite dimension case, may be denoted as <math>v^*</math> (v-star, a [[row vector]] which is the [[conjugate transpose]] to a column vector <math>v</math>).
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| In quantum mechanics, the conjugate to a ''ket vector'' <math>|\psi\rangle</math> is denoted as <math>\langle\psi|</math> – a ''bra vector'' (see [[bra-ket notation]]).
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| ==See also==
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| * [[Linear complex structure]]
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| ==References==
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| * Budinich, P. and Trautman, A. ''The Spinorial Chessboard''. Spinger-Verlag, 1988. ISBN 0-387-19078-3. (complex conjugate vector spaces are discussed in section 3.3, pag. 26).
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| [[Category:Linear algebra]]
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| [[Category:Vectors|Vector space]]
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Plastic and Reconstructive Surgeon Rolando from Blenheim, has interests including garage saleing, new launch property singapore and snorkeling. Finished a cruise ship experience that included passing by Vallée de Mai Nature Reserve.
Feel free to visit my blog post :: gamerhop.com