Infinite dimensions and matrices

[Lolsauce]

Homework Statement

Find 2 more orthonormal polynomials on the interval [-2,1] up to degree 2 given that the first polynomial p(x) = 1/√3. ( Note: Take the highest coefficient to be positive and enter your answer as a decimal.)

Homework Equations

This is a web assign equation so the answer format is in something like this, where I enter in the solutions:

degree 1 = (something)x + something
degree 2 = (something)x^2+(something)x+something

The Attempt at a Solution

I'm trying to understand this problem here. What exactly does the 1 over square root three give us? How does it help?

I was told by my professor that to do this use the dot product of two functions f and g then find integral of f(x)g(x)dx over the boundary conditions. I'm not exactly sure what this means but I followed some examples from the homeworks, as we hadn't really learned this in lecture yet.

So I make up two equations:

f(x) = a + bx
g(x) = x(a+bx)

(1) I integrate both equations on the given boundaries: Int[-2,1] (a + bx) dx= 3a - (3/2)b

(2) INT [-2,1]x(a+bx) dx = -(3/2)a + 3b

After this step I have NO IDEA what to do. I have a system of two equations. I can make a matrix
| 3 -(3/2) |
|-(3/2) 3 |

But how does this help me? If anyone could give me guidance, please do. Thank you very much

 

[Ray Vickson]

Homework Statement

Find 2 more orthonormal polynomials on the interval [-2,1] up to degree 2 given that the first polynomial p(x) = 1/√3. ( Note: Take the highest coefficient to be positive and enter your answer as a decimal.)

Homework Equations

This is a web assign equation so the answer format is in something like this, where I enter in the solutions:

degree 1 = (something)x + something
degree 2 = (something)x^2+(something)x+something

The Attempt at a Solution

I'm trying to understand this problem here. What exactly does the 1 over square root three give us? How does it help?

I was told by my professor that to do this use the dot product of two functions f and g then find integral of f(x)g(x)dx over the boundary conditions. I'm not exactly sure what this means but I followed some examples from the homeworks, as we hadn't really learned this in lecture yet.

So I make up two equations:

f(x) = a + bx
g(x) = x(a+bx)

(1) I integrate both equations on the given boundaries: Int[-2,1] (a + bx) dx= 3a - (3/2)b

(2) INT [-2,1]x(a+bx) dx = -(3/2)a + 3b

After this step I have NO IDEA what to do. I have a system of two equations. I can make a matrix
| 3 -(3/2) |
|-(3/2) 3 |

But how does this help me? If anyone could give me guidance, please do. Thank you very much

Label the three polynomials as p0(x), p1(x) and p2(x) of respective degrees 0, 1 and 2. You are told that the interval is [-2,1] and that p0(x) = 1/sqrt(3), hence int_{x=-2..1} p0(x)^2 dx = 1. That seems to be saying that the polynomials should also be "orthonormal", that is, have "squared norm" = 1. Anyway, you need int_{x=-2..1} p0(x)*p1(x) dx = 0 (p0 orthogonal to p1) and int_{x=-2..1} p1(x)^2 dx = 1 norm condition), so you can determine both constants a and b in the formula p1(x) = a + b*x. Now, with p2(x) = e + f*x +g*x^2, you can determine e, f and g from the conditions int_{x=-2..1} p0(x)*p2(x) dx = 0, int_{x=-2..1} p1(x)*p2(x) dx = 0 and int_{x=-2..1} p2(x)^2 dx = 1. You will have three simple linear equations in the three unknowns e,f and g.

RGV

 

[Lolsauce]

Label the three polynomials as p0(x), p1(x) and p2(x) of respective degrees 0, 1 and 2. You are told that the interval is [-2,1] and that p0(x) = 1/sqrt(3), hence int_{x=-2..1} p0(x)^2 dx = 1. That seems to be saying that the polynomials should also be "orthonormal", that is, have "squared norm" = 1. Anyway, you need int_{x=-2..1} p0(x)*p1(x) dx = 0 (p0 orthogonal to p1) and int_{x=-2..1} p1(x)^2 dx = 1 norm condition), so you can determine both constants a and b in the formula p1(x) = a + b*x. Now, with p2(x) = e + f*x +g*x^2, you can determine e, f and g from the conditions int_{x=-2..1} p0(x)*p2(x) dx = 0, int_{x=-2..1} p1(x)*p2(x) dx = 0 and int_{x=-2..1} p2(x)^2 dx = 1. You will have three simple linear equations in the three unknowns e,f and g.

RGV

Thank for for your response. I have a question though, I was able to solve the first degree equation of a + bx, but I can't seem to get the second one. So you set the first two equations p0p2 and p1p2 to zero correct?

 

[Lolsauce]

Never mind I got it, thanks!