how to prove that invertible matrix and vectors span the same space? Given $M$ is an invertible matrix, and {$\vec{v_1}...\vec{v_k}$} spans $R^n$,
then {A$\vec{v_1}...A\vec{v_k}$} also spans $R^n$
What does matrix invertibility have to do with span?

From scratch:
If $\vec v\in \mathbb R^n$, then so is $A^{-1}\vec v$, and so
$A^{-1}\vec v=\sum_{i=1}^{k}c_i\vec v_i$ for some $c_i\in \mathbb R$. Upon multiplying by $A$ you get 
$AA^{-1}\vec v=\vec v=A\sum_{i=1}^{k}c_i\vec v_i=\sum_{i=1}^{k}c_iA\vec v_i$.
