Prove that a positive operator is invertible Help guys, I need to prove this:
Let $(V,\langle~,~ \rangle)$ a finite n-dimensional euclidean space.Let $T$ be a linear operator defined positive (There exists a non singular operator $S$ such that $T=S^*S$) on $V$, prove that $T$ is invertible
I tried this:
We know by hypothesis that $S$ is ivertible,then there exists $S^{-1}$ such that $SS^{-1}=I$ $S$ then I need to prove that $T=T^*$
But that is trivial, since $T=S^*S=(S^*S)^*=S^*(S^*)^*=S^*S=T$
Is that correct? I need help!

$T$ is invertible if $Tx=0$ implies $x=0$. In this case, $T=S^*S$ where $S$ is non-singular. Therefore, if $Tx=0$, it follows that
$$
     0 = \langle Tx,x\rangle=\langle S^*Sx,x\rangle=\langle Sx,Sx\rangle=\|Sx\|^2 \implies Sx=0 \implies x=0.
$$
So $T$ is invertible.
