Linear Algebra, Orthonormal question

[wtmoore]

Homework Statement

I have this question that I am trying to figure out about orthonormality,I have tried to take a picture of it and put it on here but I can't figure out the url. Anyway I will try and write it out.

Show that the vector {sin(x),cos(x)} is a basis for the vector space defined by:
V={asin(x) + bcos(x) l a,b ε ℝ, 0≤ x ≤ pi} using the inner product :
<f,g>=∫(0 to pi)fgdx, fgεV
and determine an orthonormal basis.

Homework Equations

The Attempt at a Solution

I found the integral of sin(x)cos(x)dx between 0 and pi to be 0.
This make it orthonormal right as it's the same as the dot product.

Now I think I have to find out whether <f,f> is = 1 (of unit length)

so I did integral of sin^2(x)dx between 0 and pi but found pi/2. I also found the same for cos^2(x).

Does this mean they are not orthonormal? I don't know if it makes a difference that <f,f>=<g,g>.

Thanks

 

[jbunniii]

Homework Statement

I have this question that I am trying to figure out about orthonormality,I have tried to take a picture of it and put it on here but I can't figure out the url. Anyway I will try and write it out.

Show that the vector {sin(x),cos(x)} is a basis for the vector space defined by:
V={asin(x) + bcos(x) l a,b ε ℝ, 0≤ x ≤ pi} using the inner product :
<f,g>=∫(0 to pi)fgdx, fgεV
and determine an orthonormal basis.

Homework Equations

The Attempt at a Solution

I found the integral of sin(x)cos(x)dx between 0 and pi to be 0.
This make it orthonormal right as it's the same as the dot product.

Now I think I have to find out whether <f,f> is = 1 (of unit length)

so I did integral of sin^2(x)dx between 0 and pi but found pi/2. I also found the same for cos^2(x).

Does this mean they are not orthonormal? I don't know if it makes a difference that <f,f>=<g,g>.

Thanks

Correct, sin(x) and cos(x) are orthogonal [their inner product with each other is zero] but not orthonormal [their norms are not 1]. There's a very simple modification you can make to sin(x) and cos(x) which will make them orthonormal. Do you see what it is?

 

[jbunniii]

P.S. It's clear that V is spanned by sin(x) and cos(x), but the definition of a basis also requires that these functions must be linearly independent. Did you show that?

 

[genericusrnme]

What I find helps when you're first working with the notion of 'functions as vectors' is to look back to your basic 3-vectors and think about what these properties mean for them.

Orthogonal 3-vectors have a.b = 0, orthogonal functions have f.g = 0
Normalised vectors have a.a = 1, normalised functions have f.f = 1

Think about how you normalise 3-vectors. Think about how you orthogonalise 3-vectors.
Here's another little problem you can do which is kinda fun, build up a set of orthogonal polynomials using the gram-schmidt process (these polynomials are the legendre polynomials)

 

[wtmoore]

Correct, sin(x) and cos(x) are orthogonal [their inner product with each other is zero] but not orthonormal [their norms are not 1]. There's a very simple modification you can make to sin(x) and cos(x) which will make them orthonormal. Do you see what it is?

I think so, I sort of spotted it but wasn't sure. Do I define some sort of new function, let's say h(x), where h(x) = (π^-0.5)sin(x)? Then <h,h> = 1? This is normalization right?

 

[wtmoore]

P.S. It's clear that V is spanned by sin(x) and cos(x), but the definition of a basis also requires that these functions must be linearly independent. Did you show that?

I can explain it but don't know how to write this in a neat way. For example if I plug in values for x between 0 and π, let's take 0, then I know that this makes the sin term 0 and cos positive, and if we take pi/2 for example then it's the other way around. So for whatever value is taken between 0 and pi, one of them is a positive and one is 0, therefore a=b=0.

I however don't know how to express this in proper notation.

 

[jbunniii]

I think so, I sort of spotted it but wasn't sure. Do I define some sort of new function, let's say h(x), where h(x) = (π^-0.5)sin(x)? Then <h,h> = 1? This is normalization right?

Right idea, but I think your scale factor is off. If your original function had <f,f> = pi/2, then you would want to define h = sqrt(2/pi)*f.

 

[wtmoore]

Right idea, but I think your scale factor is off. If your original function had <f,f> = pi/2, then you would want to define h = sqrt(2/pi)*f.

I mean <f,f> = pi sorry, then h=pi^(-0.5)*f would work?

 

[jbunniii]

I can explain it but don't know how to write this in a neat way. For example if I plug in values for x between 0 and π, let's take 0, then I know that this makes the sin term 0 and cos positive, and if we take pi/2 for example then it's the other way around. So for whatever value is taken between 0 and pi, one of them is a positive and one is 0, therefore a=b=0.

I however don't know how to express this in proper notation.

Yes, you have the right idea. You want to show that if

a cos(x) + b sin(x) = 0 for all x,

then a and b must be 0.

So pick some specific values of x that are easy to work with. I suggest x = 0 and x = pi/2.

 

[jbunniii]

I mean <f,f> = pi sorry, then h=pi^(-0.5)*f would work?

But is <f,f> = pi?

Isn't it

\int_{0}^{\pi} sin^2(x) dx?

This should work out to pi/2, not pi. And similarly for cos(x).

 

[wtmoore]

But is <f,f> = pi?

Isn't it

\int_{0}^{\pi} sin^2(x) dx?

This should work out to pi/2, not pi. And similarly for cos(x).

Yeh originally I did it from 0 to pi but when I double checked for some reason I made a mistake and did it from -pi to pi.


Ok so I just need (pi/2)^(-1)?

 

[wtmoore]

Yes, you have the right idea. You want to show that if

a cos(x) + b sin(x) = 0 for all x,

then a and b must be 0.

So pick some specific values of x that are easy to work with. I suggest x = 0 and x = pi/2.

Yeh I tried these,

acos(0)+bsin(0)=0
Then a must be 0.

acos(pi/2)+bsin(pi/2)=0
then b must be 0.

Is it enough to write this, or is there some sort of other form I can generalize it for?

 

[jbunniii]

Yeh originally I did it from 0 to pi but when I double checked for some reason I made a mistake and did it from -pi to pi.


Ok so I just need (pi/2)^(-1)?

The square root of that.

 

[jbunniii]

Yeh I tried these,

acos(0)+bsin(0)=0
Then a must be 0.

acos(pi/2)+bsin(pi/2)=0
then b must be 0.

Is it enough to write this, or is there some sort of other form I can generalize it for?

That's enough. If the linear combination equals zero, it has to equal zero at every value of x, including the two you chose (0 and pi/2). Those two alone are enough to imply that a = b = 0. You could have picked almost any pair of values of x, as long as they resulted in a solvable system of two equations with two unknowns. 0 and pi/2 just make the math easier.

 

[wtmoore]

The square root of that.

Thanks jbunniii,

I understand all of it, and have managed to complete 2 similar questions now.

My only question is, why is it the square root? Surely if <f,f> = pi/2 then we need 2/pi to make this 1?

 

[jbunniii]

Thanks jbunniii,

I understand all of it, and have managed to complete 2 similar questions now.

My only question is, why is it the square root? Surely if <f,f> = pi/2 then we need 2/pi to make this 1?

No, suppose you set h = \alpha f. Then

&lt;h,h&gt; = &lt;\alpha f, \alpha f&gt; = \alpha^2 &lt;f,f&gt;

You want this equal to 1, so

\alpha^2 = 1/&lt;f,f&gt;

and hence

\alpha = \sqrt{1/&lt;f,f&gt;}

 

[wtmoore]

Yeh I was thinking just at the end not the fact that they are going to multiply each other. Thanks for all your help Jbunnii