TITLE: Then prove: $\vec{v} = \vec{0}$ if $\langle u,v\rangle = 0$
QUESTION [6 upvotes]: If $\vec{v} \in V$ such that $\langle u,v\rangle = 0$, $\forall \vec{u} \in V$.
Then prove: $\vec{v} = \vec{0}$
I tired to solve by assuming that they are $\langle u,v\rangle \neq 0$ $\rightarrow$ $u,v$ are not orthogonal, and if they are not orthogonal then $\langle u,v\rangle = 0$ can only be true if and only if $\vec{v} = \vec{0}$.
Is this correct? Or is there any other alternative way?

REPLY [3 votes]: If  v is a fixed vector and $\langle u,v\rangle=0$ for all u $\in V$, then in particular for $u=e_i$ for $i=1,\ldots,n$.
We know also that $v=a_1e_1+\ldots+a_ne_n$
 Then $\langle e_1,v\rangle=0=a_1\langle e_1,e_1\rangle+\ldots+a_n\langle e_1,e_n\rangle=a1\langle e_1,e_1\rangle$ 
but $\langle e_1,e_1\rangle \neq 0$ so $a_1=0$. 
Make an analogous process with all $e_i$ and you will find that $v=0$.