# Prove: Full Rank and a solution os linear system

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?

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sorry for the notation used but do not know how to express formulas in LaTeX notation in this page –  franvergara66 Aug 26 '12 at 17:44