Orthogonal Basis: Importance & Benefits

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Discussion Overview

The discussion revolves around the importance and benefits of orthogonal bases in various mathematical contexts, including linear independence, Fourier series, and the challenges of finding bases in curved spaces. The scope includes theoretical and conceptual aspects of linear algebra and functional analysis.

Discussion Character

  • Exploratory
  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • Some participants argue that orthogonal bases are crucial for applications like Fourier series, where they facilitate the representation of functions as sums of basis vectors.
  • One participant emphasizes that orthogonal basis vectors are linearly independent, meaning no vector can be expressed as a combination of others, using the example of the standard basis vectors i, j, k.
  • Another participant challenges the necessity of orthogonality, stating that in arbitrary curved spaces, it is not generally possible to find mutually orthogonal basis vectors.
  • Some participants discuss the implications of projections in relation to linear independence, questioning whether projections violate the concept of linear independence.
  • It is noted that while non-orthogonal vectors can form a basis, determining the coefficients for such vectors is more complex compared to orthogonal sets.

Areas of Agreement / Disagreement

Participants express differing views on the necessity of orthogonality for basis vectors, with some asserting its importance while others argue that it is not a requirement in all contexts, particularly in curved spaces. The discussion remains unresolved regarding the implications of projections on linear independence.

Contextual Notes

There are limitations regarding the definitions of linear independence and the conditions under which orthogonality is applicable, particularly in non-Euclidean geometries. The discussion does not resolve these complexities.

matqkks
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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

[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.
 
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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