Complex Conjugate Vector Space
In mathematics, the complex conjugate of a complex vector space V\, is a complex vector space \overline V, which has the same elements and additive group structure as V, but whose scalar multiplication involves conjugation of the scalars. In other words, the scalar multiplication of \overline V satisfies \alpha\,*\, v = where * is the scalar multiplication of \overline and \cdot is the scalar multiplication of V. The letter v stands for a vector in V, \alpha is a complex number, and \overline denotes the complex conjugate of \alpha. More concretely, the complex conjugate vector space is the same underlying vector space (same set of points, same vector addition and real scalar multiplication) with the conjugate linear complex structure J (different multiplication by i). Motivation If V and W are complex vector spaces, a function f : V \to W is antilinear if f(v + w) = f(v) + f(w) \quad \text \quad f(\alpha v) = \overline \, f(v) With the use of the conjugate vector space \over ...
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Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting poin ...
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