By conjugate linear transformation, I mean under scalar multiplication instead of C ( a

mistergoneo7

mistergoneo7

Answered question

2022-07-02

By conjugate linear transformation, I mean under scalar multiplication instead of C ( a f ) = a C ( f ), I would have C ( a f ) = a ¯ C ( f ), where a is a constant complex number, and C is the transformation.

Answer & Explanation

vrtuljakc6

vrtuljakc6

Beginner2022-07-03Added 16 answers

Given a linear-conjugate function C : V V, V a complex vector space, we define the conjugate vector space V ¯ , by being the same set V, with an altered multiplication by scalar ∗, defined by
λ x := λ ¯ x
where is V's usual multiplication. See the notations, V = ( V , + , ), and V ¯ = ( V , + , ). Reforce, the set of vectors is the same. It should be checked that V ¯ is indeed a vector space, although it's easy to see. Some facts are:
i) a set B is a basis for V iff it is a basis for V ¯ ;
ii) from i) follows that dim V = dim V ¯ ;
iii) W is a subspace of V iff it is a subspace of V ¯ ;
iv) Given a basis B, x V has coordinates ( x 1 , , x n ) B iff considering x V ¯ , we have x = ( x 1 ¯ , , x n ¯ ) B
v) C : V V is linear conjugate iff C : V ¯ V is linear iff C : V V ¯ is linear;
vi) V ¯ ¯ = V

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