Complex Conjugate Vector Space - Conjugate Linear Maps

Conjugate Linear Maps

Any linear map induces a conjugate linear map, defined by the formula

The conjugate linear map is linear. Moreover, the identity map on induces the identity map, and

for any two linear maps and . Therefore, the rules and define a functor from the category of complex vector spaces to itself.

If and are finite-dimensional and the map is described by the complex matrix with respect to the bases of and of, then the map is described by the complex conjugate of with respect to the bases of and of .