Prove linear independence of vectors transformed by linear transformation implies original vectors independent
Solution 1:
You can prove it by contradiction. Note that $0$ does not belong to any set linearly independent, so you can rule out any of the vectors being null. If $c_1v_1 + ... + c_nv_n = 0$, where some scalar is non-null, say $c_1 \neq 0$, then we can write $$ v_1 = \frac{-c_2}{c_1}v_2- \cdots\frac{-c_n}{c_1}v_n .$$
Applying $T$ $$ Tv_1 = \frac{-c_2}{c_1}Tv_2- \cdots\frac{-c_n}{c_1}Tv_n ,$$
Which Means that the set of images is linearly dependent.