Orthogonal Basis: Importance & Benefits

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SUMMARY

Orthogonal bases are crucial in various mathematical applications, particularly in Fourier series and harmonic analysis, where they ensure linear independence among basis vectors. The formula x = ∑_{i=1}^n{/ e_i} illustrates how orthogonal vectors facilitate the decomposition of functions into trigonometric series. While it is possible to have non-orthogonal bases in arbitrary curved spaces, orthogonal bases simplify the process of finding coefficients, making them preferable for linear independence in vector spaces.

PREREQUISITES
  • Understanding of Fourier series and harmonic analysis
  • Familiarity with linear algebra concepts, particularly vector spaces
  • Knowledge of linear independence and orthogonality
  • Basic proficiency in mathematical notation and summation
NEXT STEPS
  • Study the properties of orthogonal bases in linear algebra
  • Explore the application of Fourier series in signal processing
  • Learn about Gram-Schmidt orthogonalization process
  • Investigate the implications of non-orthogonal bases in curved spaces
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Mathematicians, physicists, and engineers who require a solid understanding of vector spaces, linear independence, and their applications in Fourier analysis and related fields.

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Why is an orthogonal basis important?
 
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They're important in so many ways. For example, in Fourier series, where we can say

x=\sum_{i=1}^n{\frac{<x,e_i>}{<e_i,e_i>}e_i}

And this provides the very foundation for trigonometric series and harmonic analysis.
 
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 ?
 
stallionx said:
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.
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.
 
WannabeNewton said:
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.

But one will have a projection unto another, is not this an infraction of " linear independency " ?
 
is not this an infraction of " linear independency

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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