Answer
See below
Work Step by Step
Consider:
$$\alpha_1Tv_1+\alpha_2 Tv_2+...+\alpha_n T v_n=0$$
Using linearly of $T$, we have:
$$T(\alpha_1 v_1+\alpha_2 v_2+...+\alpha_n v_n)=0\\
\rightarrow \alpha_1v_1+\alpha_2v_2+...+\alpha_n v_n \in Ker(T)$$
Since $Ker(T)=\{0\} \rightarrow \alpha_1v_1+\alpha_2v_2+...+\alpha_n v_n=0$
Since $\{v_1,v_2,...v_n\}$ is a linearly independent set, we have:
$$\alpha_1=\alpha_2=...=\alpha_n=0$$
Hence, $\{T(v_1),T(v_2),...,T(v_n)\}$ is a linearly independent subset of $W$.
