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 $b = 0$.

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

Then:

$\ds a z_1 + b z_2$

$=$

$\ds 0$

$\ds \leadstoandfrom \ \ $

$\ds a \paren {x_1 + i y_1} + b \paren {x_2 + i y_2}$

$=$

$\ds 0$

$\ds \leadstoandfrom \ \ $

$\ds a x_1 + b x_2$

$=$

$\ds 0$

$\ds a y_1 + b y_2$

$=$

$\ds 0$

$\ds \leadstoandfrom \ \ $

$\ds a x_1$

$=$

$\ds -b x_2$

$\ds a y_1$

$=$

$\ds -b y_2$

$\ds \leadstoandfrom \ \ $

$\ds \dfrac {y_1} {x_1}$

$=$

$\ds \dfrac {y_2} {x_2}$

and $z_1$ and $z_2$ are parallel.

$\blacksquare$

1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Solved Problems: Graphical Representations of Complex Numbers. Vectors: $9$