In mathematics, a mappingf : V → W from a complex vector space to another is said to be antilinear (or conjugate-linear or semilinear) if Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... In mathematics and related technical fields, the term map or mapping is often a synonym for function. ... Vector space - Wikipedia, the free encyclopedia /**/ @import /skins-1. ...
for all a, b in C and all x, y in V. The composition of two antilinear maps is complex-linear.
An antilinear map may be equivalently described in terms of linear map to the complex conjugate vector space. 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...
See also: complex conjugate, sesquilinear form In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ... In mathematics, a sesquilinear form on a complex vector space V is a map V × V → C that is linear in one argument and conjugate-linear in the other. ...
In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication.
Furthermore, in the case that V=W, this vector space is an associative algebra under composition of maps, since the composition of two linear maps is again a linear map, and the composition of maps is always associative.
Given again the finite dimensional case, if bases have been chosen, then the composition of linear maps corresponds to the matrix multiplication, the addition of linear maps corresponds to the matrix addition, and the multiplication of linear maps with scalars corresponds to the multiplication of matrices with scalars.
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