In mathematics, one associates to every complex vector space V its complex conjugate vector space V*, again a complex vector space. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added The underlying set of vectors and the addition of V* are the same as those of V, and the scalar multiplication in V* is defined as follows:
- 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. In Mathematics, the complex conjugate of a Complex number is given by changing the sign of the Imaginary part.
The map * : V → V* defined by x* = x for all x in V is then bijective and antilinear. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property In Mathematics, a mapping f: V → W from a Complex vector space to another is said to be antilinear (or conjugate-linear Furthermore, we have V** = V and x** = x for all x in V.
Given any other bijective antilinear map from V to some vector space W, we can show that W and V* are isomorphic as complex vector spaces. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective
Given a linear map f : V → W, the conjugate linear map f* : V* → W* is defined as follows:
- f * (x * ) = f(x) * . In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that
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 branch of Mathematics, a functor is a special type of mapping between categories In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets
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*. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally Basis vector redirects here For basis vector in the context of crystals see Crystal structure.
Note that V and V* have the same dimension over C and are therefore isomorphic as C vector spaces. In Mathematics, the dimension of a Vector space V is the cardinality (i However, there is no natural isomorphism from V to V*. In Category theory, a branch of Mathematics, a natural transformation provides a way of transforming one Functor into another while respecting the internal
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