- #1
ainster31
- 158
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Why is the math in the red box necessary? According to this definition, it isn't:
ainster31 said:Why is the math in the red box necessary? According to this definition, it isn't:
tiny-tim said:hi ainster31!
sorry, i don't understand your question …
the red box proves that (φ0, φn) = 0 (for n ≠ 0)
dextercioby said:m=0 is contained as a particular case for arbitrary m and n. It's no need to make the particular case.
dextercioby said:The proof goes directly by putting cos a = Re (e^ia).
ainster31 said:According to definition 12.1.3, a set of real-valued functions can be proven to be orthogonal if (φm, φn) = 0. So why is it necessary to prove (φ0, φn) = 0?
ainster31 said:So you're saying it was unnecessary?[...]
Orthogonality refers to the mathematical concept of two or more objects being perpendicular or at right angles to each other. In the context of functions, orthogonality means that the inner product (or dot product) of any two functions in the set is equal to zero.
Proving that a set of functions is orthogonal is important because it allows us to determine if the functions are linearly independent. This is a crucial concept in many areas of mathematics, such as in solving differential equations, Fourier analysis, and linear algebra.
The process for proving a set of functions is orthogonal involves using the definition of orthogonality and the properties of inner products to show that the inner product of any two functions in the set is equal to zero. This typically involves using integration techniques and algebraic manipulations.
Some common techniques used in proving a set of functions is orthogonal include the use of trigonometric identities, substitution, integration by parts, and using known orthogonal functions (such as the Legendre polynomials or the trigonometric functions) to help simplify the inner product.
Yes, it is possible to prove that a set of functions is orthogonal without using integration. This can be done by using the definition of orthogonality and properties of inner products, along with algebraic manipulations, to show that the inner product of any two functions in the set is equal to zero. However, integration is often a useful tool in proving orthogonality and may make the process easier.