The image of any linearly independent set under whatever linear transformation is lineraly independent
Solution 1:
Your hypothesis only says that $a_i=0$ if $\sum_{i=1}^m a_i s_i = 0$. But for a (general) linear map $T$, $T(v)$ can be zero even if $v \neq 0$. To make your argument work, you need $T$ to be injective so that $T(v)=0 \implies v=0$. And under this extra condition, $T(S)$ is indeed linearly independent.