Orthonormal Basis

**Swlabr** · January 13th 2010, 03:04 AM

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).

**Defunkt** · January 13th 2010, 04:54 AM

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,

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

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

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