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Let $T_A : \mathbb R^n \to \mathbb R^m$ be given by the matrix $\mathbf{A}$. I have been told A has a left inverse if $T_A$ is injective. Also, I have been told that $\mathbf{A}$ has a left inverse if $(A^TA)^{-1} = I$.

Therefore, I was thinking that $(A^TA)^{-1} = I$ must imply that $T_A$ is injective. If this is true, can anyone give an explanation of this implication?

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  • $\begingroup$ A weaker condition: assume that $A^TA$ is invertible, i.e. $(A^TA)^{-1}A^TA=I$. That's all you need to make a conclusion that $A$ has a left inverse. $\endgroup$ – A.Γ. Aug 16 at 22:16
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Suppose $(A^T A)^{-1} = I$. Then $A^T A = I$. Hence $A$ has a left inverse ($A^T$). Therefore, $T_A$ is injective.

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