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

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.

• Are you trying to show that they are linearly dependent? Since $v_1 + v_2 = v_1 + v_2$, there's no way that these vectors can be linearly independent. – platty Dec 13 '18 at 0:24

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