Left Invertibility and Injectivity

A matrix $A \in \mathbb{R}^{m \times n}$ is left invertible if there exists $L \in \mathbb{R}^{n \times m}$ such that $LA = I_n$. Composing $L$ on the left of $A$ gives the identity.

The connection to injectivity: $A$ is left invertible if and only if $A$ is injective. Proof: $LA = I$ means for any $\mathbf{x}$ with $A\mathbf{x} = \mathbf{0}$, we get $\mathbf{x} = I\mathbf{x} = (LA)\mathbf{x} = L(A\mathbf{x}) = L\mathbf{0} = \mathbf{0}$. So the null space is trivial.

The left inverse "undoes" $A$ — if $A\mathbf{x} = \mathbf{b}$, then $\mathbf{x} = L\mathbf{b}$. This only works because $A$ is injective: each $\mathbf{b}$ has at most one pre-image, so the formula is unambiguous.