# Math Help - Orthonormal Basis

1. ## Orthonormal Basis

Let $\{u, v, w\}$ be an orthonormal basis of $\mathbb{R}^3$. Then is it true that every element $\alpha$ of this vector space can be written thus,

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

If so, why?

( $\alpha.u$ etc. is the dot product).

2. Yes, this is correct for any finite-dimensional vector space V (with some inner product) over $\mathbb{F = R} ~or~ \mathbb{F = C}$.

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

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

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

Therefore: $v = \sum_{i=1}^n c_iv_i = \sum_{i=1}^n v_i$