Complex Vector Space

linear-algebra

Definition

Complex Vector Space

A complex vector space is a vector space whose underlying field of scalars is the field of complex numbers $\mathbb{C}$. Equivalently, it is a set $V$ with vector addition and scalar multiplication

$$\mathbb{C}\times V\to V$$

satisfying the usual axioms of a vector space.

Example: The space

$${\mathbb{C}}^n=\{(z_1,\dots ,z_n)\mid z_i\in \mathbb{C}\}$$

is a complex vector space with componentwise addition and scalar multiplication.