Complex Conjugate Vector Space
In mathematics, the (formal) complex conjugate of a complex vector space is the complex vector space consisting of all formal complex conjugates of elements of . That is, is a vector space whose elements are in one-to-one correspondence with the elements of :
with the following rules for addition and scalar multiplication:
Here and are vectors in, is a complex number, and denotes the complex conjugate of .
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).
In the case where is a linear subspace of, the formal complex conjugate is naturally isomorphic to the actual complex conjugate subspace of in .
Read more about Complex Conjugate Vector Space: Antilinear Maps, Conjugate Linear Maps, Structure of The Conjugate, Complex Conjugate of A Hilbert Space
Famous quotes containing the words complex and/or space:
“Young children constantly invent new explanations to account for complex processes. And since their inventions change from week to week, furnishing the “correct” explanation is not quite so important as conveying a willingness to discuss the subject. Become an “askable parent.””
—Ruth Formanek (20th century)
“The merit of those who fill a space in the world’s history, who are borne forward, as it were, by the weight of thousands whom they lead, shed a perfume less sweet than do the sacrifices of private virtue.”
—Ralph Waldo Emerson (1803–1882)