Inner Product: Equation for x^p & x^q | 65 Chars

In summary, the conversation discusses finding an equation for the inner product of x^p and x^q, given that <f,g>= \int_0^1 f(x)g(x)x^2 dx for f(x)= x^p and g(x)= x^q. It is concluded that the inner product can be expressed as <f,g>= \int_0^1 x^{p+q+2}dx, and an exact number can be given for this expression.
  • #1
keddelove
3
0
Hello, I am working on an assignment were I have shown that a certain equation defines an inner product, which was simple enough. Te equation was:

[tex] \left\langle {f,g} \right\rangle = \int\limits_0^1 {f\left( x \right)g\left( x \right)x^2 dx} [/tex]My question then is: How do i state an equation for the inner product of x^p and x^q.

Sorry if the information is sparse
 
Last edited:
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  • #2
You set f(x)=x^p and g(x)=x^q.
 
  • #3
Looks straight forward to me. If
[tex]<f,g>= \int_0^1 f(x)g(x)x^2 dx[/tex]
f(x)= xp, and g(x)= xq, then
[tex]<f,g>= \int_0^1 (x^p)(x^q)x^2dx= \int_0^1 x^{p+q+2}dx[/tex]
 
  • #4
HallsofIvy said:
Looks straight forward to me. If
[tex]<f,g>= \int_0^1 f(x)g(x)x^2 dx[/tex]
f(x)= xp, and g(x)= xq, then
[tex]<f,g>= \int_0^1 (x^p)(x^q)x^2dx= \int_0^1 x^{p+q+2}dx[/tex]
And you can give an exact number equal to that last expression.
 

Related to Inner Product: Equation for x^p & x^q | 65 Chars

1. What is the definition of an inner product?

An inner product is a mathematical operation that takes in two vectors and produces a scalar value. It is used to measure the angle between two vectors and the length of a vector.

2. How is the inner product calculated?

The inner product is calculated by taking the dot product of the two vectors and then multiplying it by the cosine of the angle between them.

3. What is the equation for x^p & x^q?

The equation for x^p & x^q is ⟨x^p, x^q⟩ = ∑(x_i)^p * (x_i)^q, where x_i represents the components of the two vectors.

4. What is the significance of the exponent in the inner product equation?

The exponent in the inner product equation represents the order of the vector. It determines the type of inner product being calculated, such as the L2 norm (p = 2) or the L1 norm (p = 1).

5. How is the inner product used in real-world applications?

The inner product is used in various fields of science and engineering, such as computer vision, signal processing, and quantum mechanics. It is also used in statistics and machine learning for dimensionality reduction and feature extraction.

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