Orthogonality of functions.

waynex · #1 $Old$ November 4th, 2009, 08:01 AM

Hi there,

I have a question that is as follows,

Find values of a,b,c,d,e for which the set of functions
$1, a+bx, c + dx +ex^2$ is orthogonal on $C([-1,1])$

I'm not really sure on where to start with this one and I was hoping someone could help.

Thanks for all replies in advance,

Wayne.

robeuler · #2 $Old$ November 4th, 2009, 08:30 AM

Quote:

Originally Posted by waynex $View Post$

Hi there,

I have a question that is as follows,

Find values of a,b,c,d,e for which the set of functions
$1, a+bx, c + dx +ex^2$ is orthogonal on $C([-1,1])$

I'm not really sure on where to start with this one and I was hoping someone could help.

Thanks for all replies in advance,

Wayne.

Do you know what the innerproduct on C([-1,1]) is?

A set is orthogonal if and only the elements have pairwise innerproduct=0

Opalg · #3 $Old$ November 4th, 2009, 08:40 AM

Quote:

Originally Posted by waynex $View Post$

Hi there,

I have a question that is as follows,

Find values of a,b,c,d,e for which the set of functions
$1, a+bx, c + dx +ex^2$ is orthogonal on $C([-1,1])$

I'm not really sure on where to start with this one and I was hoping someone could help.

The condition for f(x) and g(x) to be orthogonal is $\int_{-1}^1f(x)g(x)\,dx = 0$ . So the condition for 1 and a+bx to be orthogonal is $\int_{-1}^1(a+bx)\,dx = 0$ . Choose values for a and b that satisfy that condition. Then write down the conditions for $c + dx +ex^2$ to be orthogonal to 1 and also orthogonal to a+bx. That will give you two conditions that c, d and e must satisfy.

An alternative method is to start with the set of functions $\{1, x, x^2\}$ , and apply the Gram–Schmidt procedure to it. But obviously you can only do that if you have learned about the Gram–Schmidt process.

waynex · #4 $Old$ November 4th, 2009, 08:43 AM

Thanks for that, I was unsure about whether I needed to use Gram-Schmidt or not. I guess my old head isn't working today. Thanks again for your quick replies.