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...