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Math Help - [SOLVED] Gram-Schmidt

  1. #1
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    [SOLVED] Gram-Schmidt

    the vectors {a1, a2, a3} are orthonormal
    the vector a4 is a combination x1a1 + x2a2 + x3a3

    after applying gram schmidt to the set of vectors {a1, a2, a3, a4} the resulting set of vectors will be:

    {a1, a2, a3, zero vector}
    is this correct?
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  2. #2
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    Yes, if you do it in that order- that is, normalize a_1 to unit length, then "Gram-Schmidt" a_2, etc. Since the first three are already orthonormal, you would just get a_1, a_2, a_3 again. And since a_3 is a linear combination of them, it has no "part" that is perpendicular to the space spanned by them- Gram-Schmidt would give the 0 vector.

    But if you started by normalizing a_3, then projecting, say, a_1 onto that to find a vector perpendicular to a_3 you would get 3 completely different vectors.
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