Linear Dependence Lemma 2 I'm trying to interpret the very last sentence. He says nothing about $a_{1}$, so in the case of $a_{1}=0$ all the other coefficients can not be zero, which would imply linearly independence. The case where $a_{1}\neq 0$ all the other coefficients can not be zero because, if they were, we would have $a_{1}v_{1}=0$ which is never true since $v_{1}\neq 0$.
Is this correct? Thank you in advance.
I have seen this post post about lin.dep. lemma., but I think my question is unanswered in that post.


What you've said is perfectly correct. A slightly quicker/neater/different way to see it (without splitting into cases) might be the following:
Suppose instead that all of $a_2,\dots,a_m$ are equal to zero. Then $a_1v_1=0$. But $v_1\not=0$ so it must be that $a_1=0$, contradicting our original assumption that not all of the $a_i$ are $0$.
