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Let $V$ and $W$ be finite dimensional inner product spaces, $T:V\to W$ an injective linear transformation and $T^*:W\to V$ its adjoint, i.e. the linear transformation satisfying: \begin{equation} \langle Tv,w\rangle=\langle v,T^*w\rangle \end{equation} for all $v\in V$ and $w\in W$. Is $T^*T:V\to V$ necessarily an isomorphism? It is injective because if $T^*Tv=0$ then $\langle T^*Tv,v\rangle=\langle 0,v\rangle=0$ then $\langle Tv,Tv\rangle=0$ and then $v=0$. But I'm not sure if it's surjective.

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If $T^*Tv=0$, then $$ 0=\langle v,T^*Tv\rangle=\langle Tv,Tv\rangle $$ so $Tv=0$ and, being $T$ injective, $v=0$ (which is essentially what you did).

Now an injective endomorphism of a finite dimensional vector space is necessarily surjective (consequence of the rank-nullity theorem).

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