Answer
See answer below
Work Step by Step
Assume that $[v_1,v_2,...,v_n]$ is linearly independent (1)
and $A$ is an invertible $n \times x$ matrix.
Let $c_1,c_2,...c_n$ be scalars such that
$c_1Av_1+c_2Av_2+...+c_kAv_k=0$
$\rightarrow A(c_1v_1+c_2v_2+...+c_kv_k)=0$
Since $A$ is invertible, the equation $Ax=0$ has only one solution.
$\rightarrow c_1v_1+c_2v_2+...+c_kv_k=0$ (2)
From (1) and (2) $\rightarrow c_1=c_2=...c_k=0$
Hence, the set $[Av_1, Av_2,...Av_k]$ is linearly independent.
