Register to reply

Orthogonal basis

Share this thread:
matqkks
#1
Aug27-11, 08:00 AM
P: 153
Why is an orthogonal basis important?
Phys.Org News Partner Mathematics news on Phys.org
Heat distributions help researchers to understand curved space
Professor quantifies how 'one thing leads to another'
Team announces construction of a formal computer-verified proof of the Kepler conjecture
micromass
#2
Aug27-11, 08:36 AM
Mentor
micromass's Avatar
P: 18,327
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
#3
Aug27-11, 09:05 AM
P: 100
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
#4
Aug27-11, 10:28 AM
C. Spirit
Sci Advisor
Thanks
WannabeNewton's Avatar
P: 5,635
Orthogonal basis

Quote Quote by stallionx View Post
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.
stallionx
#5
Aug27-11, 01:32 PM
P: 100
Quote Quote by WannabeNewton View Post
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 " ?
Studiot
#6
Aug27-11, 03:21 PM
P: 5,462
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.


Register to reply

Related Discussions
Orthogonal Basis Calculus & Beyond Homework 4
Normalising orthogonal basis Calculus & Beyond Homework 6
Difference between an orthogonal complement and the orthogonal complement's basis. Calculus & Beyond Homework 2
Inner products and orthogonal basis Linear & Abstract Algebra 4
Orthogonal basis Linear & Abstract Algebra 29