I have been trying to solve the following system of linear equations in the complex plane:
$$\begin{cases} z_1 = -iz_2 \\ z_2 = iz_1 \end{cases} $$
I know the solution, it's $z_1 = 1, \space z_2 = i$, but i can't find a way to prove it, it seems like the solution is "hidden". If someone could provide a proof (and maybe an explanation on what's going on), I would much appreciate it.
Best Answer
Hint
$e^{i\frac{\pi}2}=i\Rightarrow\frac 1i=e^{-i\frac{\pi}2}=-i\rightarrow\boxed{\frac 1i=-i}$