I recently encountered a problem in Hoffmann-Kunze linear algebra:
If $(.,.)$ is the standard inner product on $\mathbb C^2$ then show that $(Tv,v)=0 \forall v\in \mathbb C^2 \implies T=0$, I think the proof is quite tricky and I saw its solution at last after trying myself through a long time. The solution is like this, we put $x+y$ and $x+iy$ for $v$ and prove that $(Tx,y)=0 \forall x,y\in \mathbb C^2$, thus it follows that $Tx=0 \forall x\in \mathbb C^2$, but I think it cannot be just a trick, there might be a deeper understanding within this simple 'just a trick' looking proof. I want to visualize why this fact is not true for $\mathbb R^2$ space but true when $\mathbb C$ replaces $\mathbb R$.
I am looking for an intuitive solution of the same problem or a suitable visualization that would help me in better understanding of this problem.