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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.2012-12-04
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    What is $a$? Is it the same thing as $A$?2012-12-04
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    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.2014-12-13

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