How Do You Construct an Orthonormal System from Functions?

[overseastar]

Hey,
I've taken linear algebra a long ago, and now came across a simple-looking question that i need help with

Construct an orthonormal system from the three functions:
1, x, 3x^2 - 1

Can anyone give me a pointer to solving this question?
Thanks a bunch!

 

[Muzza]

Apply Gram-Schmidt?

 

[overseastar]

oh ya, this question is from a calculus homework before we get into Fourier Series and integral, so I don't think it wants us to apply Gram-Schmidt method.

 

[matt grime]

ortho, and normal, only make sense with respect to an inner product. You haven't given one. And just because you've not been taught the formal proof of Gram-Schmidt doesn't mean you're not supposed to figure it out on your own; it is merely projection formally written down and not at all a tricky thing to figure out on your own.

 

[overseastar]

it was with respect to 1...

 

[matt grime]

1 what?
I can think of many different inner products on function spaces, and polynomial rings, none of them is called 1.

 

[overseastar]

Sorry...Here's the entire question

Show that 1, x, 3x^2 - 1 are orthogonal functions with respect to the weight function 1 on the interval [-1, 1]. Construct an orthonormal system from the three functions.

I got the first part of the question, I'm stuck on the orthnormal part.

Currently looking through linear algebra notes, not quite understanding how to do it using Gram-Schmidt.

 

[matt grime]

You know how to do projection? Ie given vectors v and w write w as the sum of a vector parallel to v and one orthogonal to v. That's all gram schmidt is, but you just do it again, and again, and again...

 

[overseastar]

I thought Gram-Schmidt is used to find orthogonal basis only.
Sorry I might sound dumb, but the last time I took linear algebra was 2 years ago...don't remember much from it.

 

[matt grime]

You can't get from a set of orthogonal vectors to a set of orthogonal vectors all of length one?

 

[overseastar]

how do we do that?

 

[matt grime]

If I were to say here are u and v two orthogonal vectors (none of which is zero), you can now write down two orthogonal vectors of length 1, right? I can't tell if you're kidding me because you think I"m being patronizing or if you can't find a vector of length one from another vector (it has been some years you say since you did this).

 

[overseastar]

I'm not a very wordy person and I learn from seeing equations and numbers and examples, I guess it's kinda hard to explain it like that. Thanks for your help, I'll think about it for now.

according to my notes, it says that a system of orthogonal functions w.r.t. weight q of [a, b] is also orthonormal if (see attachment) for all m. Is that what you meant?

 

[matt grime]

If v is a vector (not the zero vector) then v/|v| is a unit vector in the same direction.

 

[overseastar]

So if i were to rewrite the 3 functions in terms of vectors, would they become like this?

0 0 1 <-- 1
0 1 0 <-- x
3 0 -1 <-- 3x^2-1

so 1st vector would be
0
0
3
?

 

[overseastar]

If v is a vector (not the zero vector) then v/|v| is a unit vector in the same direction.

Ok, I finally understand the part after the unit vector.
What is the next step once I've got the unit vector for each of the function?