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?

Actually, the opposite is true: the problem in your question is half of the proof of the proposition in the title, namely, that $\{Av_1,\dotsc,Av_k\}$ is a spanning set whenever $\{v_1,\dotsc,v_k\}$ is. So, let $x \in \mathbb{R}^n$. Then, of course, $A^{-1}x \in \mathbb{R}^n$, so $A^{-1}x = \sum_{i=1}^k \alpha_i v_i$ for some $\alpha_i \in \mathbb{R}$ -- note that I'm only using the fact that $\{v_1,\dotsc,v_k\}$ is a spanning set, and am assuming absolutely nothing concerning linear independence. Since $x = A(A^{-1}x)$, what follows?
