How to prove orthogonality on a set of functions?

[Jay9313]

Homework Statement

A set of functions, F, is given below. Determine the size of the largest subset of F which is mutually orthogonal on the interval [-1, 1], and find all such subsets of this size. Show all of your work.

F = { 1, x, x² , sin(x), cos(x), cosh(x), sinh(x)}

Homework Equations

Not really any relevant equations, but I am aware that for a function to be orthogonal on symmetric bounds, the integral of the product of the functions on those symmetric bounds must be equal to 0.

The Attempt at a Solution

I have yet to complete an attempt because I'm not quite sure how to start. Any help would be appreciated.

 

[Dick]

Homework Statement

A set of functions, F, is given below. Determine the size of the largest subset of F which is mutually orthogonal on the interval [-1, 1], and find all such subsets of this size. Show all of your work.

F = { 1, x, x² , sin(x), cos(x), cosh(x), sinh(x)}

Homework Equations

Not really any relevant equations, but I am aware that for a function to be orthogonal on symmetric bounds, the integral of the product of the functions on those symmetric bounds must be equal to 0.

The Attempt at a Solution

I have yet to complete an attempt because I'm not quite sure how to start. Any help would be appreciated.

Start by finding some pairs of functions in the set that are orthogonal. Since you are working on a symmetric interval considering whether functions are even or odd can help a lot. You really don't have to do much integrating.

 

[Jay9313]

Start by finding some pairs of functions in the set that are orthogonal. Since you are working on a symmetric interval considering whether functions are even or odd can help a lot. You really don't have to do much integrating.

Yeah, I got that! Lol. I'm just wondering, because it seems that the largest subset can only contain two functions..

 

[Dick]

Yeah, I got that! Lol. I'm just wondering, because it seems that the largest subset can only contain two functions..

Seems that way to me too. Can you explain why?

 

[Jay9313]

Oh, I was confused. I thought that if you multiply all of the functions together, then they will end up being an odd function, meaning that the entire set multiplied together is mutually orthogonal. BUT the question was asking about two functions at a time being multiplied together. It makes a lot more sense. Now I just have to prove which functions you can and can't multiply together to form a subset. Thank you!

 

[Dick]

Oh, I was confused. I thought that if you multiply all of the functions together, then they will end up being an odd function, meaning that the entire set multiplied together is mutually orthogonal. BUT the question was asking about two functions at a time being multiplied together. It makes a lot more sense. Now I just have to prove which functions you can and can't multiply together to form a subset. Thank you!

You are getting there. But you can have a set of more than two mutually orthogonal functions. For example, {sin(x),sin(2x),sin(3x)} are mutually orthogonal on the interval [-2pi,2pi]. Any pair you pick out of that set are orthogonal. For your set and interval, that doesn't happen. Any set of three functions won't be mutually orthogonal. You just have to figure out the reason.