How to find orthonormal basis for inner product space? In $\mathbb{R}^3$ we declare an inner product as follows: $\langle v,u \rangle \:=\:v^t\begin{pmatrix}1 & 0 & 0 \\0 & 2 & 0 \\0 & 0 & 3\end{pmatrix}u$  
How can I find an orthonormal basis for this inner product space using the Gram–Schmidt process?

Three steps which will always result in an orthonormal basis for $\mathbb R^n$:


*

*Take a basis $\{w_1,w_2,\dots,w_n\}$ for $\mathbb R^n$ (any basis is good)

*Orthogonalize the basis (using gramm-schmidt), resulting in a orthogonal basis $\{v_1,v_2,\dots,v_n\}$ for $\mathbb R^n$

*Normalize the vectors $v_i$ to obtain $u_i=\frac{v_i}{||v_i||}$ which form a orthonormal basis.

