Complex conjugate of a vector space

In mathematics, one associates to every complex vector space V its complex conjugate vector space V^*, again a complex vector space. 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.

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.

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.

Given a linear map f : V → W, the conjugate linear map f^* : V^* → W^* is defined as follows:

$f^{*}(x^{*})=f(x)^{*}.$

As you may verify for yourself, f^* is a linear map and * becomes a functor from the category of C-vector spaces to itself.

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^*.

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^*.