How to show that $v_1+v_2,v_1,v_2,\dots,v_n$ are linearly independent in a $K$-vector space $(V,*,+)$?

Question

How to show that $v_1+v_2,v_1,v_2,\dots,v_n$ are linearly independent in a $K$-vector space $(V,*,+)$, given that $v_1,v_2,v_3,\dots,v_n$ are linearly independent?

My attempt:

We know that vectors $a_1,\dots,a_n$ are linearly independent if $\lambda_1 a_1 + \lambda_2 a_2 + \dots + \lambda_n a_n=0$. But how can I show that with $v_1+v_2,v_1,v_2,v_3,\dots,v_n$?

I don't have a clue how to show this. My problem is that this is so general.

Answer 1

let $\lambda_1 = 1$, let $\lambda_2 = -1$, let $\lambda_3 = -1$. For all other let $\lambda_k = 0$