Are all $n\times n$ invertible matrices change-of-coordinates matrices in $\mathbb{R}^n$? More precisely, I'm trying to prove the following problem:
Assume that $\text{Span}\{\vec{v}_{1},\dots,\vec{v}_{k}\}=\mathbb{R}^{n}$
  and that A
  is an invertible matrix. Prove that $\text{Span}\{A\vec{v}_{1},\dots,A\vec{v}_{k}\}=\mathbb{R}^{n}$.
but I think I need the lemma in the title. I find it difficult to prove without large amounts of handwaving. Is the proposition in the title true, and if so, how to prove it?

Let $v$ be a general vector and set $w = A^{-1} v$.  Write $w = \displaystyle\sum_i c_i v_i$ since $\{v_i\}$ is a spanning set.  Then $v = A w =  A \displaystyle\sum_i c_i v_i = \displaystyle\sum_i c_i (A v_i)$ and so we can write any vector $v$ as a combination of the vectors $\{A v_i\}$
