- #1

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Why is an orthogonal basis important?

- Thread starter matqkks
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- #1

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Why is an orthogonal basis important?

- #2

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[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.

- #3

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Take i , j , k

can you solve for a ,b ,c in ai+bj+ck = 0 without setting all to zero ?

- #4

WannabeNewton

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They don'tBecause 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.

- #5

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But one will have a projection unto another, is not this an infraction of " linear independency " ?They don'thaveto be orthogonal. In an arbitrary curved space, it is not generally possible to find basis vectors that are mutually orthogonal.

- #6

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No it is not an infraction.is not this an infraction of " linear independency

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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