Proposition. If $u$ and $v$ are nonezero and $\left\langle u,v\right\rangle=0$ then $u$ and $v$ are linearly independent.
Proof. Suppose $\alpha u+\beta v=0$. Then,
(The statement is my question) $0=\left\langle \alpha u+\beta v,u\right\rangle=\alpha\left\langle u,u \right\rangle + \beta \left\langle v,v \right \rangle =\alpha \left\| u\right\| ^2$. As $u\neq 0$, $\left\| u\right\|^2\neq 0$. So $\alpha =0$.
Similarly, $\beta =0$.
My questios is the statement, how did we obtain these equivalents?