Practice Inner Product And Orthogonality In Function Spaces (27.12)

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6 sample questions

Question 1.
Define the inner product of two functions.
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Think about the integral of the product of the functions over the given interval.
Question 2.
What does orthogonality mean for two functions?
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Recall the definition of inner product in relation to zero.
3Question 3.
What is the definition of the inner product of two functions?
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Recap the definition we talked about.
4Question 4.
Are sin(x) and cos(x) orthogonal over the interval [0, 2π]?
- True
- False
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Think about their integral product.
Question 5.
Prove that the functions f(x) = x and g(x) = x^2 are not orthogonal over the interval [-1, 1].
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Remember to use the odd property of the function x^3 over symmetric limits.
Question 6.
Construct the Fourier series for a square wave using orthogonal sine functions over the interval [0, T].
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Think about how coefficients are calculated using the integral of the square wave function multiplied by the sine functions.

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