In mathematics, the complexification of a vector space V over the real number field is the 'corresponding' vector space V^C over the complex number field. That is, it shares the same dimension as V; and a basis for V over the real numbers can serve as a basis for V^C over the complex numbers.

For example, if V consists of the m×n matrices with real coefficients, V^C would consist of m×n complex matrices.

For the sake of having a basis-free definition, one can take

,

the tensor product over the real field of V and the complex numbers.