Orthogonal basis

matqkks

Why is an orthogonal basis important?

micromass

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.

stallionx

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 ?

WannabeNewton

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.

stallionx

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

Studiot

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.