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Suppose $V$, $W$ are inner product spaces, and $V$ is finite-dimensional. I need to prove that if $T \in \mathcal{L}(M)$ is an injective linear map, and $T^*$ is the adjoint of $T$, then $T^*T$ is invertible.

I got as far as to say that if $T$ is injective, then $T^*$ is surjective. But I don't know how to show that $T^*T$ is invertible. Showing that $T^*T$ is surjective or injective would imply invertibility, but I'm not sure how to do that either. I was hoping to find a way to show that $T^*$ is injective (which would then imply that $T^*T$ is injective) but I wasn't able to.

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    $\begingroup$ What is $M$? What is the domain and range of $T$? $\endgroup$ – irchans Aug 14 at 1:37
  • $\begingroup$ $T^*Tx = 0$ implies $\langle Tx, Tx \rangle = \langle T^* Tx,x \rangle = 0$ implies $Tx = 0$ implies $x = 0$. So, $T^*T$ injective. $\endgroup$ – mathworker21 Aug 14 at 2:16
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Hint: Show that $T^*Tx = 0$ implies that $Tx = 0$, which means that $T^*T$ is also injective.

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