Does any know how to go about proving the following statement?
Let $v_1, v_2, \dots, v_n \in V$ be a linearly independent vectors. Furthermore, let $T \in \mathcal{L}(V)$ be an invertible linear transformation. Prove that $Tv_1, Tv_2, \dots, Tv_n$ are linearly independent.
Clearly we need to show that the equation
$$ c_1Tv_1 + c_2Tv_2 + \cdots + c_nTv_n = 0 $$
has only the trivial solution. We know that the equation
$$ a_1v_1 + a_2v_2 + \cdots + a_nv_n = 0 $$
has only the trivial solution. Can we use this, with the linearly of $T$, to prove the claim?