Find a g=a+bx that is orthagonal to the constant function

In summary, in the real linear space C(1, e), an inner product is defined by the equation (f,g) = integral(1 to e)(logx)f(g)g(x)dx. The norm of f(x)=sqrt(x) is ll f ll = 1/2sqrt(e2+1). For part (b), the linear polynomial g(x)=a+bx is required to be orthogonal to the constant function f(x)=1, meaning that (f, g) = 0. Applying the definition of the inner product, we can solve for conditions on a and b, which leads to the solution b[x-(e2+1)/4].
  • #1
Cassi
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Homework Statement


In the real linear space C(1, e), define an inner product by the equation (f,g) = integral(1 to e)(logx)f(g)g(x)dx.
(a) If f(x)=sqrt(x), compute ll f ll (the norm of f)
(b) Find a linear polynomial g(x)=a+bx that is orthagonal to the constant function f(x)=1

Homework Equations



The Attempt at a Solution


I have solved part (a) and found that ll f ll = 1/2sqrt(e2+1) but I am having trouble with part B.

I see that the answer is b[x-(e2+1)/4] but I cannot get this answer. I know that (f, g) = 0 but I do not know how to solve this.
 
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  • #2
Cassi said:
I know that (f, g) = 0 but I do not know how to solve this.
Did you apply the definition of the inner product to this equation? Then you can solve it and get conditions for a and b.
 

FAQ: Find a g=a+bx that is orthagonal to the constant function

1. What is the meaning of "orthagonal" in this context?

In mathematics, two functions are considered orthogonal if their inner product (integral of the product of the two functions) is equal to zero. In simpler terms, this means that the two functions are perpendicular to each other on a graph.

2. How do you find the value of g in this equation?

To find the value of g, you will need to use the formula g = a + bx, where a and b are constants and x is the independent variable. You can determine the values of a and b by solving a system of equations using the orthogonality condition (integral of g multiplied by the constant function is equal to zero).

3. Can you provide an example of finding a g that is orthogonal to the constant function?

Sure, let's say we have the constant function f(x) = 2 and we want to find a g that is orthogonal to it. We can start by setting up the integral of g multiplied by f(x) equal to zero. This gives us the equation ∫g(x)2dx = 0. We can then solve for g(x) by finding the antiderivative of 2, which is 2x. Therefore, g(x) = 2x + c, where c is a constant. This satisfies the orthogonality condition and g(x) is orthogonal to the constant function f(x) = 2.

4. What is the significance of finding a g that is orthogonal to the constant function?

Finding a g that is orthogonal to the constant function is useful for various applications in mathematics and physics. It allows us to solve problems involving orthogonal bases, such as Fourier series and eigenfunctions. It also helps in finding solutions to differential equations and in approximating functions with simpler functions.

5. Are there any other conditions for finding a g that is orthogonal to the constant function?

Yes, there are other conditions that must be satisfied in order for g to be orthogonal to the constant function. One important condition is that g must be continuous and square-integrable (the integral of g squared must be finite). This ensures that the inner product between g and the constant function is well-defined. Additionally, g must also be orthogonal to all other functions in the orthogonal basis being used.

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