I have read from many sources the well-known result that an orthonormal list of vectors $v_1,\ldots, v_m$ is linearly independent. The most common procedure is to first assume that there exist scalars $a_1,\ldots, a_m$ such that $a_1v_1 + \cdots + a_mv_m = 0$.
I'm probably overlooking something basic, but can someone explain how the existence of such a set of scalars is guaranteed?
The definition of linear independence says that $v_1, \dots, v_m$ are linearly independent if for any $c_1, \dots, c_m \in \mathbb{F}$, $$c_1v_1 + \dots + c_mv_m = 0 \implies c_1 = \dots = c_m = 0.$$ So one way to prove linear independence is to assume $c_1v_1 + \dots + c_mv_m = 0$ and then show that $c_1 = \dots = c_m = 0$.