# Orthonormal Basis

• Jan 13th 2010, 03:04 AM
Swlabr
Orthonormal Basis
Let $\displaystyle \{u, v, w\}$ be an orthonormal basis of $\displaystyle \mathbb{R}^3$. Then is it true that every element $\displaystyle \alpha$ of this vector space can be written thus,

$\displaystyle \alpha = (\alpha.u)u+(\alpha.v)v+(\alpha.w)w$.

If so, why?

($\displaystyle \alpha.u$ etc. is the dot product).
• Jan 13th 2010, 04:54 AM
Defunkt
Yes, this is correct for any finite-dimensional vector space V (with some inner product) over $\displaystyle \mathbb{F = R} ~or~ \mathbb{F = C}$.

Assume $\displaystyle \{v_1,v_2,...,v_n\} \subset V$ is an orthonormal basis of $\displaystyle V$.

Let $\displaystyle v \in V \Rightarrow v = \sum_{i=1}^n c_iv_i$ for some scalars $\displaystyle c_i \in \mathbb{F}$. But then,

$\displaystyle \forall 1 \leq j \leq n, ~ <v,v_j> = <\sum_{i=1}^n c_iv_i,v_j> \overbrace{=}^{Orthonormality} c_j<v_j,v_j> = c_j||v_j||^2 =$ $\displaystyle c_j \Rightarrow \forall 1 \leq j \leq n, ~ \boxed{c_j = <v,v_j>}$

Therefore: $\displaystyle v = \sum_{i=1}^n c_iv_i = \sum_{i=1}^n <v,v_i>v_i$