Lesson plan
Real structure
In mathematics, a real structure on a complex vector space is a way to decompose the complex vector space in the direct sum of two real vector spaces. The prototype of such a structure is the field of complex numbers itself, considered as a complex vector space over itself and with the conjugation map \sigma: {\mathbb C} \to {\mathbb C}\, , with \sigma (z)={\bar z} , giving the "canonical" real structure on {\mathbb C}\, , that is {\mathbb C}={\mathbb R}\oplus i{\mathbb R}\, . The conjugation map is antilinear: \sigma (\lambda z)={\bar \lambda}\sigma(z)\, and \sigma (z_1+z_2)=\sigma(z_1)+\sigma(z_2)\, .
Article summary
In mathematics, a real structure on a complex vector space is a way to decompose the complex vector space in the direct sum of two real vector spaces. The prototype of such a structure is the field of complex numbers itself, considered as a complex vector space over itself and with the conjugation map \sigma: {\mathbb C} \to {\mathbb C}\, , with \sigma (z)={\bar z} , giving the "canonical" real structure on {\mathbb C}\, , that is {\mathbb C}={\mathbb R}\oplus i{\mathbb R}\, . The conjugation map is antilinear: \sigma (\lambda z)={\bar \lambda}\sigma(z)\, and \sigma (z_1+z_2)=\sigma(z_1)+\sigma(z_2)\, .
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Discussion questions
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Bibliography
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