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Author Archives: hecker
Linear Algebra and Its Applications, Exercise 3.4.26
Exercise 3.4.26. In the Gram-Schmidt orthogonalization process the third component is computed as . Verify that is orthogonal to both and . Answer: Taking the dot product of and we have Since and are scalars and and are orthonormal we … Continue reading
Linear Algebra and Its Applications, Exercise 3.4.25
Exercise 3.4.25. Given over the interval what is the closest line to the parabola formed by ? Answer: This amounts to finding a least-squares solution to the equation , where the entries 1, , and are understood as functions of … Continue reading
Linear Algebra and Its Applications, Exercise 3.4.24
Exercise 3.4.24. As discussed on page 178, the first three Legendre polynomials are 1, , and . Find the next Legendre polynomial; it will be a cubic polynomial defined for and will be orthogonal to the first three Legendre polynomials. … Continue reading
Linear Algebra and Its Applications, Exercise 3.4.23
Exercise 3.4.23. Given the step function with for and for , find the following Fourier coefficients: Answer: For the numerator is and the denominator is so that . For the numerator is so that . For the numerator is and … Continue reading
Linear Algebra and Its Applications, Exercise 3.4.22
Exercise 3.4.22. Given an arbitrary function find the coefficient that minimizes the quantity (Use the method of setting the derivative to zero.) How does this value of compare with the Fourier coefficient ? What is if ? Answer: We are … Continue reading
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Linear Algebra and Its Applications, Exercise 3.4.21
Exercise 3.4.21. Given the function on the interval , what is the closest function to ? What is the closest line to ? Answer: To find the closest function to the function we first project onto the function on the … Continue reading
Linear Algebra and Its Applications, Exercise 3.4.20
Exercise 3.4.20. Given the vector what is the length ? Given the function for what is the length of the function over the interval? Given the function for what is the inner product of and ? Answer: We have so … Continue reading
Linear Algebra and Its Applications, Exercise 3.4.19
Exercise 3.4.19. When doing Gram-Schmidt orthogonalization, an alternative approach to computing (equation 7 on page 173) is to instead compute in two separate steps: Show that the second method is equivalent to the first. Answer: We substitute the expression for … Continue reading
Linear Algebra and Its Applications, Exercise 3.4.18
Exercise 3.4.18. If is the projection matrix onto the column space of the matrix and , what is a simple formula for ? Answer: The projection matrix onto the column space of can be calculated as . Since the columns … Continue reading
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Tagged orthogonalization, orthonormal vectors, projection matrix
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Linear Algebra and Its Applications, Exercise 3.4.17
Exercise 3.4.17. Given the matrix from the previous exercise and the vector , solve by least squares using the factorization . Answer: From the previous exercise we have To find the least squares solution to where , we take advantage … Continue reading
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