complexification
Sign in to saveIn mathematics, the complexification of a vector space over the field of real numbers (a "real vector space") yields a vector space over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include their scaling ("multiplication") by complex numbers. Any basis for (a space over the real numbers) may also serve as a basis for over the complex numbers.
~9 min read
Article
9 sectionsContents
- Formal definition
- Basic properties
- Examples
- Dickson doubling
- Complex conjugation
- Linear transformations
- Dual spaces and tensor products
- See also
- References
In mathematics, the complexification of a vector space over the field of real numbers (a "real vector space") yields a vector space over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include their scaling ("multiplication") by complex numbers. Any basis for (a space over the real numbers) may also serve as a basis for over the complex numbers.
== Formal definition == Let V be a real vector space. The ' of is defined by taking the tensor product of V with the complex numbers (thought of as a 2-dimensional vector space over the reals):