# If $T:V\to W$ is an injective linear transformation and $T^*:W\to V$ is its adjoint, is $T^*T:V\to V$ necessarily an isomorphism

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: $$\langle Tv,w\rangle=\langle v,T^*w\rangle$$ 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.

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).