Actually, you wrote things backward. Why did you take $a_1Tv_1+\dots+a_nTv_n=0$ with $a_1,\dots,a_n=0$? Actually, I can really take a linearly dependent family of vectors $v_1=v_2=\dots=v_n=v$ and say "take $a_1v_1+\dots+a_nv_n=0$ with $a_1=\dots=a_n=0$. Then the family is independent because $a_i=0$".
The casual way you prove $P\implies Q$ is that you assume $P$ is true and prove $Q$.
You want to prove that
$Tv_1,\dots,Tv_n$ are linearly independent $\implies$ $v_1,\dots,v_n$ are linearly independent.
So, let's go. Assume that $Tv_1,\dots,Tv_n$ are linearly independent. You tried to translate what it means, which is a good thing, but used it in a quite wrong way. Your assumption means that:
If $a_1,\dots,a_n$ are real numbers such that $a_1Tv_1+\dots+a_nTv_n=0$, then $a_1=\dots=a_n=0$.
This is your assumption. This is a hypothesis. You have nothing to use it at for the moment.
Now what's your goal? You want to show that $v_1,\dots,v_n$ are linearly independent. Translate this:
If $a_1,\dots,a_n$ are real numbers such that $a_1v_1+\dots+a_nv_n=0$, then $a_1=\dots=a_n=0$.
So let's prove this. Let $a_1,\dots,a_n$ be real numbers such that $a_1v_1+\dots+a_nv_n=0$. Why is $a_1=\dots=a_n=0$? I'll let you find out.