The following is a fact in Murphy's C*-algebras and operator theory page 49:
Suppose $u,v \in B(H)$, where $H$ is a Hilbert space, then $u=v$ if and only if $\langle u\xi,\xi\rangle = \langle v\xi,\xi\rangle$ for all $\xi \in H$.
Clearly if $u=v$ then $\langle u\xi,\xi\rangle = \langle v\xi,\xi\rangle$ for all $\xi\in H$. For converse, I should show $\langle u\xi,\eta\rangle=\langle v\xi,\eta\rangle$ for all $\xi,\eta\in H$, while I can not show it. I think about Polarisation identity but I can not use it. Please help me. Thanks in advance.