I'm studying for my exam of linear algebra.. I want to prove the following corollary:
Given $A \in{R^{n\times n}}$, there is a solution $x$ to $Ax = y$ for all $y$, if and only if $A$ has rank $m$ (full row rank).
I know that the rank of a matrix is the maximum number of columns (rows respectively) that are linearly independent and is defined by:
$\operatorname{Img} (A) = \operatorname{Rg} (A):= y \in{C^m}:y = Ax, x \in{C^n}$
My problem is that I can not find a way to relate the two concepts in order to reach a formal proof. Any help?