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Math Help - Change of basis matrices

  1. #1
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    Change of basis matrices

    Prove that the change of basis matrix from one orthonormal basis of \mathbb{R}_{n \times 1} to another is always orthogonal.
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  2. #2
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    If \{u_i\} and \{v_i\} are bases for the same vector space, then A is the "change of basis matrix" if and only if Au_i= v_i and u_i= A^{-1}v_i. If \{v_i\} is an orthonormal basis then <v_i, v_j}= \delta_{ij} where \delta_{ij} is the "Kronecker delta", equal to 1 if i= j, 0 otherwise.

    <v_i, v_j>= <Au_i, Au_j>= <u_i, A^TAu_j>= \delta_{ij}
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