Theorem. A linear operator $T$ on an inner product space is $0$ if and only if $\langle a,T(a)\rangle =0$ for all $a$.
(Ref: Hassani, Sadri. Mathematical physics: a modern introduction to its foundations. Springer Science & Business Media, 2013.)
Ok, let's take $\mathbb{R}^2$ with its usual inner product; $$\langle x,y \rangle :=x^\top y,\ \forall x,y\in \mathbb{R}^2.$$
Now, consider the linear transformation $$T(x):=[x_2,-x_1]^\top,$$ which satisfies $$ \langle x,T(x) \rangle=x_1x_2-x_2x_1=0,\ \forall x\in \mathbb{R}^2. $$ But, $T$ is not a zero transformation since, it does not map every vector $x$ to $0$. Where is my mistake?