Linear Combination of Non-Parallel Complex Numbers is Zero if Factors are Both Zero

Theorem
Let $z_1$ and $z_2$ be complex numbers expressed as vectors such taht $z_1$ is not parallel to $z_2$.

Let $a, b \in \R$ be real numbers such that:
 * $a z_1 + b z_2 = 0$

Then $a = 0$ and $b = 0$.

Proof
Suppose it is not the case that $a = b = 0$.

Then:

and $z_1$ and $z_2$ are parallel.