$\mathbb{F}$ is $\mathbb{C}$ or $\mathbb{R}$.
given $u\in\mathbb{F}^n$ such as $\langle v,u \rangle=0$ for each $v\in\mathbb{F}^n$.
prove that $u=\underline{0}$.
by contradiction, assume that $u \neq \underline{0}$. now, because $v$ and $u$ are both in $\mathbb{F}^n$ I can choose $v=u$ so: $\langle v,u \rangle = \langle u,u \rangle = 0$. and from positivity this means that $u=\underline{0}$.
Is this correct? I'm not sure because I learn this as part of diagonalization and don't see how this is connected to the topic.
Thanks in advance.
No contradiction is needed: $$ 0 = u \cdot u = \lVert u \rVert^2 = \sum_i \left|u_i\right|^2 \Rightarrow u_i = 0 \Rightarrow u = 0 $$