What is the formula for a line in $\mathbb{C}^d$?

Question

I know the formula for a line in $\mathbb{C}$. But i wonder that what is the formula for a line in $\mathbb{C}^d$ where d is just a positive integer greater than 1 if we dont know anything about this line? I mean it is just an arbitrary line. How many equation do we need? Can you give an example? Thank you.

Answer 1

Pick 2 elements $a,b\in\mathbb C^d$ with $a\neq 0$. Then a line is all elements of the form $ax+b$; a complex line if $x\in\mathbb C$, a real line if $x\in\mathbb R$.

The lines are not uniquely determined by $a$ and $b$.

Can you give an example?

$$\ell = \ell(x) = \binom{1+i}{2}\cdot x \;+\; \binom{i\pi}{\sqrt{2}} \in \mathbb C^2$$