Prove the linear independence of $\{v_1,v_2+v_2 \}\,$? 
Prove that if $\{v_1,v_2\}$ are linearly independent (LI), then also $\{v_1,v_1+v_2\}$ are linearly independent.
Is the converse true?

I have proved the first part, but now i am stuck on whether the converse is true.
that is, if the $ \{ v_1,v_1+v_2 \} $ are LI, so are $\{v_1,v_2\}$.
How can i prove this?
Thanks in advance.

The reverse can be proved with the usual definition of linear independence.
Let $c_1,c_2\in\mathbb{R}$ be such that $c_1 v_1+c_2v_2=0$. Then
$$0=c_1v_1+c_2v_2=c_1v_1+c_2v_2+c_2v_1-c_2v_1=(c_1-c_2)v_1+c_2(v_1+v_2)$$
Because $v_1,v_1+v_2$ are linearly independent, we must conclude $c_1-c_2=c_2=0$, which leads to $c_1=c_2=0$ and proves $v_1,v_2$ are linearly independent.
