Solution to the following complex system

Question

I have to solve the following system:

$(−3i)x − y = 0$

$9x − 3iy = 0$

My professor claims that the solution is: $t(i, 3) $, while I get:$t(1,-3i)$

Answer 1

And you are both right! You won't agree on which $t$ gives rise to any particular solution, but your parametrisation describes exactly the same solution set as your professor's.

For instance, your professor thinks that $(2, 6i)$ is a solution because that's what he gets when he sets $t = -2i$. You think it's a solution because that's what you get when you set $t = 2$. You don't agree on the $t$, but you agree that it is a solution.

Answer 2

The set of solutions coincide, using the reparametrization $t=-it'$, i.e., $$ \{t(i,3)\mid t\in \Bbb C\}=\{ t'(1,-3i)\mid t'\in \Bbb C\}. $$