# Definition A point in $\mathbb{C}^n$ is a sequence of $n$ complex numbers $(z^1, z^2, \ldots, z^n)$ called an n-tuples of complex numbers. # Vector Space $\mathbb{C}^n$ forms a [[vector space]] where addition and scalar multiplication have their usual definitions. Note that, unlike $\mathbb{R^n}$ ([[Real Coordinate Space|R^n]]), $\mathbb{C^n}$ can be either a complex vector space (i.e. defined over the field $\mathbb{F} = \mathbb{C}$), or a real vector space ($\mathbb{F} = \mathbb{R}$, in which case it is often called $\mathbb{C}^n_{\mathbb{R}}$). # Dimension Consider $\mathbb{C}$, it clearly has dimension $1$ since you can take, e.g., $(1)$ as the basis, and you can obtain any $z \in \mathbb{C}$ by multiplying it by the "basis vector" $1$. Thus, $\mathbb{C}$ is one-dimensional. However, when we consider $\mathbb{C}$ as a real vector space (i.e. $\mathbb{C}_\mathbb{R}$), it has dimension $2$, since you can only obtain a complex number from real numbers by multiplication by the set $\{1, i\}$. So is $\mathbb{C}$ one or two dimensional? It has *complex dimension* $1$ and *real dimension* $2$, i.e., the dimension depends on the [[field]] the vector space is defined over. Similarly, $\mathbb{C}^n$ has *complex dimension* $n$ and *real dimension* $2n$. ## Applications in Physics In physics, $\mathbb{C}^n$ most often occurs as the [[Ket Space|ket space]] of finite dimensional quantum mechanical systems, e.g., particles with [[Physics/Spin]] and no translational [[degree of freedom|degrees of freedom]]. For a particle with spin $s$, its ket space is $\mathbb{C}^{2s+1}$.