How to write the equation of a line in $\mathbb C^n$? I want to write the equation of a line in $\mathbb C^n$ passing through a point $(z_1,z_2,...,z_n)$. Actually I have a set of points and I suspect they all lie on the same line which passes through this point and I want a convenient way to check it.
Thank you
 A: Since $\mathbb{C}$ is $2$-dimensional, as a real vector space, a one-dimensional subspace of $\mathbb{C}^n$ is actually a complex plane, and is therefore not an ordinary line.
However, such subspaces/"lines" can be expressed as in the real case, as
a solution to $n-1$ equations of the form $\lambda_1 z_1+\lambda_2 z_2+ \cdots+ \lambda_n z_n = c$ where c is $0$ if you want a subspace, any complex number if you want any set that "looks" like $\mathbb{C}$.
Edit: Fixed error.
A: How much do you know about vectors?  You can take two points and find the difference between them, as a vector; this is the analogue of the slope of a line in $\mathbb{R}^2$.  If the vector is $\mathbf{v}$ and one point is $a$, then points on the line are of the form $a+\lambda\mathbf{v}$, where $\lambda$ is real.  (Unless you're talking about a "complex line," which is really a plane.)
Also, if you're just uncomfortable with $\mathbb{C}^n$, you can just rewrite all your points as points in $\mathbb{R}^{2n}$, like $(Re(z_1),Im(z_1),\dotsc,Re(z_n),Im(z_n))$.  This is an isomorphism of the real vector space structure of $\mathbb{C}^n$, so in particular it preserves lines and addition.  The only thing you lose is complex multiplication.
EDIT: If you're looking for a complex line, just let $\lambda$ be complex in the above vector expression.
A: It doesn't matter if you work with complex, real numbers, or elements of any field $\mathbb{K}$: if you have a point $p = (z_1, \dots , z_n) \in \mathbb{K}^n$, or any $\mathbb{K}$-vector space $V$, an equation for a straight line in $\mathbb{K}^n$ (or in $V$) passing through $p$ may always be written, for instance, as
$$
p + \lambda v \ ,
$$
with $v= (v_1, \dots , v_n) \in \mathbb{K}^n$ (or $v\in V$) and $\lambda \in \mathbb{K}$.
