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Math Help - Orthonormal Basis

  1. #1
    MHF Contributor Swlabr's Avatar
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    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).
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  2. #2
    Super Member
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    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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