- #1
roto25
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How would you prove, using the integral product, that the set of {cos x, cos 2x, cos 3x, cos 4x, ...} is an orthogonal set?
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Is the Set {cos x, cos 2x, cos 3x, ...} Orthogonal Using Integral Products?
- Thread starter roto25
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In summary, to prove that the set of {cos x, cos 2x, cos 3x, cos 4x, ...} is an orthogonal set, we can use the integral product and show that the integral of the product of any two elements from the set is equal to zero, satisfying the condition of orthogonality. This can be done by setting up the integral and using integration by parts.
- #2
tiny-tim
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welcome to pf!
hi roto25! welcome to pf!
i] define the integral product
ii] define orthogonal set
iii] apply i and ii …
hi roto25! welcome to pf!
i] define the integral product
ii] define orthogonal set
iii] apply i and ii …
what do you get?
- #3
Bavid
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over the interval -pi to pi, the integral of cos(mx)cos(nx)dx is zero, as long as m and n are integers. Therefore, if you select ANY pair of elements from the set, the 'integral of their product' will be zero, thereby satisfying the condition of orthogonality.
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chiro
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Bavid said:
On top of what Bavid said if you don't know where to start set up the integral and use integration by parts.
- #5
blue_raver22
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To prove that a set is orthogonal, we need to show that the inner product of any two distinct elements in the set is equal to 0. In this case, we are dealing with a set of trigonometric functions {cos x, cos 2x, cos 3x, cos 4x, ...}.
The inner product of two functions f(x) and g(x) is defined as:
∫f(x)g(x)dx
To prove that the set is orthogonal, we need to show that for any two distinct elements of the set, the inner product is equal to 0. Let's take two arbitrary elements from the set, cos nx and cos mx, where n and m are distinct positive integers.
∫cos nx cos mx dx
= ∫cos nx cos mx dx
= ∫cos nx cos mx dx
= ∫cos nx cos mx dx (using the trigonometric identity cos(A+B)=cosAcosB-sinAsinB)
= 0 (since n and m are distinct positive integers, nx and mx will never be equal, so cos nx cos mx will always be 0)
Therefore, the inner product of any two distinct elements in the set is equal to 0, proving that the set {cos x, cos 2x, cos 3x, cos 4x, ...} is an orthogonal set.
FAQ: Is the Set {cos x, cos 2x, cos 3x, ...} Orthogonal Using Integral Products?
What does it mean for a set to be orthogonal?
A set is considered orthogonal if all of its elements are orthogonal to each other. In other words, the inner product of any two elements in the set is equal to 0.
How do you prove that a set is orthogonal?
To prove that a set is orthogonal, you must show that the inner product of any two elements in the set is equal to 0. This can be done through various methods such as using the definition of orthogonality, the Gram-Schmidt process, or the Pythagorean theorem.
Can a set contain both orthogonal and non-orthogonal elements?
No, a set cannot contain both orthogonal and non-orthogonal elements. If even one element in the set is not orthogonal to another element, then the entire set is not considered orthogonal.
Is orthogonality the same as independence?
No, orthogonality and independence are not the same. While a set of orthogonal vectors is always linearly independent, a set of linearly independent vectors is not always orthogonal. However, in certain cases, orthogonality can imply independence.
Why is it important to prove that a set is orthogonal?
Proving that a set is orthogonal is important in various areas of mathematics and science, such as linear algebra, signal processing, and quantum mechanics. Orthogonal sets have useful properties that make calculations and analyses easier, and they also allow for the simplification of complex systems and models.