Really stuck computing orthogonal complement?

[physman55]

Really stuck... computing orthogonal complement?

Homework Statement

The Attempt at a Solution

I'm really sorry I can't provide much here because I really don't know how to proceed. Could anyone offer a hint to get me started?

 

[lurflurf]

You have a Legendre polynomial hint. Which polynomials of degree 5 are orthogonal to all polynomials of degree 3?

 

[physman55]

The Legendre polynomials are...

 

[lurflurf]

Legendre polynomials

The polynomials given
P0(x) = 1
P1(x) = x
P2(x) = ( 3 x^2 - 1 ) / 2
P3(x) = ( 5 x^3 - 3 x ) / 2
P4(x) = ( 35 x^4 - 30 x^2 + 3 ) / 8
P5(x) = ( 63 x^5 - 70 x^3 +15) / 8

for which
<Pi,Pj>=0 whenever i!=j

 

[physman55]

Sorry I'm not really understanding what you're saying. Aren't each of the legendre polynomial Pi's orthogonal to each of the basis vectors for P3 {1,x,x^2,x^3}? Isn't that what we're looking for?

 

[lurflurf]

Yes that is what we are looking for.
P4(x) = ( 35 x^4 - 30 x^2 + 3 ) / 8
and
P5(x) = ( 63 x^5 - 70 x^3 +15) / 8
are orthogonal to all lower polynomials {1,x,x^2,x^3} and are thus a basis for the orthogonal complement.

 

[physman55]

Hmmm actually I take that back... the LP polynomials are orth. w.r.t. other LP polynomials, but not w.r.t. the standard basis for polynomials of degree at most 3. :s

 

[lurflurf]

Legendre polynomials are orthogonal to all polynomials of lower order.
Suppose
p3(R)=span{1,x,x^2,x^3}=span{P0,P1,P2,P3}
<P4,p3(R)>=0
<P5,p3(R)>=0

 

[physman55]

Right, gotcha. Thanks!