Suppose that $\{v_{1}\cdot\cdot\cdot v_{n}\}$ are unit vectors in $\mathbb{R}^{n}$ such that $$\ \|v\|^{2}=\sum_{1}^{n} |\langle v | v_{i} \rangle|^{2}, \forall v\in\mathbb{R}^{n}.$$ Then how to prove that the vectors $v_{1}\cdot\cdot\cdot v_{n}$ are mutually orthogonal?
Please don't mind it may be very simple for some one. I tired it but could't find.
We have to prove that $\langle v_{i}|v_{j}\rangle =0.$
One more thing i know is that if $v$ is any vector that is orthogonal to each of $v_{i}$ then by given relation we have $\ \|v\|=0$ and so $v=0.$
Please tell me how to show that the vectors $v_{1}\cdot\cdot\cdot v_{n}$ are mutually orthogonal? Thanks in advance.
$4<x,y>= \vert\vert x + y\vert\vert ^2- \vert\vert x - y\vert\vert ^2$
Thus from your formula $4<x,y>= \sum _1^n \vert < x + y, e_i >\vert ^2- < x - y, e_i >\vert ^2$ and $<x,y>= \sum _1^n <x,e_i><y,e_i>= <x,\sum _1^n<y,e_i> e_i>$
By substraction $y-\sum _1^n<y,e_i> e_i$ is orthogonal to every $x$, and is therefore $0$.
It follows that the $n$ vector $e_i$ generate $\bf R^n$ : it is a base, and the coordinates of $y$ in this base are $<y,e_i>$, in particular $<e_i,e_j>= \delta _i^j$, and this is an orthonormal basis.