Prop. Let $V$ be a vectos space over $K$ and let $v_1,...,v_n$ be linearly independt. Then, if
$\alpha_1 v_1+...+\alpha_n v_n=\beta_1 v_1+...+\beta_n v_n$
then
$\alpha_1=\beta_1,...,\alpha_n=\beta_n$
Proof. Suppose $\alpha_1 v_1+...+\alpha_n v_n=\beta_1 v_1+...+\beta_n v_n$. Then $(\alpha_1-\beta_1)v_1+...+(\alpha_n-\beta_n)v_n=0$
So, by assumption $\alpha_i - \beta_i=0$ for $i=1,...,n.$
My question is: Which assumption $\alpha_i - \beta_i=0$ for $i=1,...,n$?