Orthogonal basis

Why is an orthogonal basis important?

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 ?

WannabeNewton
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.

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.