Understanding a proof of propositon of linearly independent.

Question

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 $i=1,...,n$?

Answer 1

Definition of linear independence: $v_{1},...,v_{n}$ are said to be linearly dependent if and only if $\lambda_{1}v_{1} +...+\lambda_{n}v_{n} = 0 \Rightarrow \lambda_{1}=\lambda_{2} =...=\lambda_{n}=0$

This is what you have assumed, and then your proof follows.