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Orthogonal basis

  1. Aug 27, 2011 #1
    Why is an orthogonal basis important?
     
  2. jcsd
  3. Aug 27, 2011 #2

    micromass

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    They're important in so many ways. For example, in Fourier series, where we can say

    [tex]x=\sum_{i=1}^n{\frac{<x,e_i>}{<e_i,e_i>}e_i}[/tex]

    And this provides the very foundation for trigonometric series and harmonic analysis.
     
  4. Aug 27, 2011 #3
    Because basis vectors have got to be orthogonal ( perpendicular ) so that they are Linearly Independent and one of them can not be formed from any combo of others.

    Take i , j , k

    can you solve for a ,b ,c in ai+bj+ck = 0 without setting all to zero ?
     
  5. Aug 27, 2011 #4

    WannabeNewton

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    They don't have to be orthogonal. In an arbitrary curved space, it is not generally possible to find basis vectors that are mutually orthogonal.
     
  6. Aug 27, 2011 #5
    But one will have a projection unto another, is not this an infraction of " linear independency " ?
     
  7. Aug 27, 2011 #6
    No it is not an infraction.

    Any set of enough non parallel vectors from a vector space can be used as a basis.
    However finding the correct coefficients is more difficult (laborious) than for an orthogonal set since the orthogonality means they can be found one at a time.
     
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