how can I prove the left pseudo-inverse identity?

How can I prove that the left pseudo-inverse of matrix $A$ is $$A^{\dagger}_l = (A^TA)^{-1}A^T,$$

I know that this is true only if $\mathrm{rank}(A)=n$. and that if $\mathrm{rank}(A)=n$ that means that $A^TA$ is invertible.

I also proved that $A^{\dagger}_lA =I$

I tried to play with multiplying the matrices but it didn't help...

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What's your definition of pseudo inverse? One possible definition is exactly the formula you want to prove. – Dominik Dec 4 '12 at 20:14
What is $a$? Is it the same thing as $A$? – Chris Eagle Dec 4 '12 at 21:32
So... you proved it right? I don't think I understand what you are asking, since it seems like you proved what you are trying to prove. – Braindead Dec 13 '14 at 16:21

Actually $A^{\dagger}_l = (A^TA)^{-1}A^T$ is just A left inverse of $A$, $(A^TMA)^{-1}A^TM$ is also the left inverse for any matrix $M$ such that $A^TMA$ is invertible.