A left inverse of $A\in\mathbb F^{m\times n}$ is a matrix $L\in\mathbb F^{n\times m}$ with $LA=I_n$. It exists exactly when $A$ has full column rank $n$, requiring $m\ge n$. Over $\mathbb R$ or $\mathbb C$, one choice is $L=(A^*A)^{-1}A^*$.
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