Answer
See below
Work Step by Step
We are given $\{v_1,v_2...,v_k\}$ is a linearly independet set of vectors in $V$.
Let $\alpha_1,\alpha_2,...,\alpha_k$ be scarlars such as:
$$\alpha_1v_1+\alpha_2v_2+....+a_kv_k=0 \\
\rightarrow \alpha_1=\alpha_2=...=\alpha_k=0$$ (1)
Let:
$$\alpha_1 T(v_1),\alpha_2T(v_2),...,\alpha_kT(v_k)=0 \\
T(\alpha_1(v_1)+\alpha_2(v_2)+...+\alpha_kv(v_k))=0 \\
\alpha_1v_1+\alpha_2v_2+...+a_kv_k=0$$
From (1)
$$\alpha_1=\alpha_2=....=\alpha_k=0$$
$(T(v_1),T(v_2),....,T(v_k)\}$ is a linearly independent set.
