Proof that T*T is non-negative on linear transformations

Question

I've been doing a few problems and ran upon one I've tried to solve with no luck:

Let $T: V \to W$ be a linear transformation and $T^*$ be its adjoint, prove that $T^*T$ is non-negative.

Most of the things I've tried have just gotten me more confused and nowhere near a valid proof.

Answer 1

$T^*$ has the propety that: $$\langle T(v),w \rangle =\langle v,T^*(w)\rangle,\;\; \forall v,w\in V$$ I remember that a non-negative linear trasformation $L:V\to V$ is such that: $$\langle L(v),v\rangle\geq 0,\;\; \forall v \in V $$ in our case, we have: $$\langle v,T^*(T(v)) \rangle=\langle T(v),T(v) \rangle\geq 0$$ where i use the positivity of the inner product.