Complex N-Vectors (C^n)

We've worked with $n$-dimensional vectors in $\mathbb{R}^n$. 

What if we allow the entries of an $n$-dimensional vector to be complex? That is:

$$\vec{v} = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix}$$

where each $v_i \in \mathbb{C}$, i.e., each $v_i$ is a complex number.

Note: This includes the case where the $v_i$ are real since real numbers are a subset of complex numbers $\left( \mathbb{R} \subset \mathbb{C}\right)$.

For example:

$$\begin{bmatrix} 1 - 3i \\ 1 + 2i \end{bmatrix}, \quad \begin{bmatrix} 1 + 3i \\ 2 - \sqrt{2}i \\ 2 \end{bmatrix}, \quad \text{etc...}$$

No problem! We can let the scalars that we multiply the vectors by also be complex  as well i.e., $c\vec{v}$, where $c \in \mathbb{C}$.

All our ideas around addition, subtraction, and scalar multiplication that we've learned for $\mathbb{R}^n$ carry over.

We call the set of all $n$-dimensional vectors with complex entries $\mathbb{C}^n$.

So:

$$...$$