In mathematics, one associates to every vector space V over the complex numbers C its complex conjugate vector space V^*, again a vector space over C. The underlying set and the addition of V^* are the same as those of V, and the scalar multiplication in V^* is defined as follows: Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ...

to multiply the complex number α with the vector x in V^*, take the complex conjugate α^* of α and multiply it with x in the original space V.

The map * : V → V^* defined by x^* = x for all x in V is then bijective and antilinear. Furthermore, we have V^** = V and x^** = x for all x in V. In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ... In mathematics, a real linear transformation f from a complex vector space V to another is said to be antilinear (or conjugate-linear or semilinear) if for all c, d in C and all x, y in V. See also: complex conjugate, sesquilinear form ...

Given any other bijective antilinear map from V to some vector space W, we can show that W and V^* are isomorphic as C-vector spaces. In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ...

Given a linear map f : V → W, the conjugate linear map f^* : V^* → W^* is defined as follows: In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...

f ^* (x ^* ) = f(x) ^* .

As you may verify for yourself, f^* is a linear map and * becomes a functor from the category of C-vector spaces to itself. In category theory, a functor is a special type of mapping between categories. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ...

If V and W are finite-dimensional and the map f is described by the matrix A with respect to the bases B of V and C of W, then the map f^* is described by the complex conjugate of A with respect to the bases B^* of V^* and C^* of W^*. For the square matrix section, see square matrix. ... In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V...

Note that V and V^* have the same dimension over C and are therefore isomorphic as C vector spaces. However, there is no natural isomorphism from V to V^*. In mathematics, the dimension of a vector space V is the cardinality (i. ... In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ...