Definition. Let $x_{1},...,x_{n}$ be vectors. We say that $x_{1},...,x_{n}$ are linearly independent if for all scalars $\alpha _{1},...,\alpha _{n}$
$\alpha _{1}x_{1}+\ldots +\alpha _{n} x_{n}=0$
implies $\alpha _{1}=...=\alpha _{n}=0$.
Simple case. If $x_{1},x_{2}$ are not dependent, then $\alpha _{1},\alpha _{2}$ such that
$\alpha _{1}x_{1}+\alpha _{2}x_{2}=0$
and either $\alpha _{1} \neq 0$ or $\alpha _{2} \neq 0$. Say $\alpha _{1} \neq 0$. Then,
$x_{1}=-\dfrac {\alpha _{2}} {\alpha _{1}} x_{2}$.
My question is: The definitio says that $\alpha _{1}=...=\alpha _{n}=0$. Yet, the simple case says that either $\alpha _{1} \neq 0$ or $\alpha _{2} \neq 0$. I didn't understand. Can you explain?