In a finite dimensional inner product space with $T ∈ L(V)$, show that $\langle u,v\rangle = \langle T(u),T(v)\rangle$ implies $T$ is invertible.

Question

Here is how I've tried to go about it, and I'm curious if it's true or if I'm way off base.

T is invertible iff null$(T)=\{0\}$. Let $v∈V$ and suppose $T(v)=0$. If we can show that $v=0$, then $T$ is invertible.

Consider $\langle v,v\rangle = \langle T(v),T(v)\rangle = \langle 0,0 \rangle = 0$, and $\langle v,v\rangle = 0$ iff $v=0$.

Does this prove the statement? Or are there any other thoughts or methods?

Thanks in advance!