Author Archives: hecker

Linear Algebra and Its Applications, Exercise 3.4.28

Exercise 3.4.28. Given the plane and the following vectors in the plane, find an orthonormal basis for the subspace represented by the plane. Report the dimension of the subspace and the number of nonzero vectors produced by Gram-Schmidt orthogonalization. Answer: … Continue reading

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Linear Algebra and Its Applications, Exercise 3.4.27

Exercise 3.4.27. Given the subspace spanned by the three vectors find vectors , , and that form an orthonormal basis for the subspace. Answer: We can save some time by noting that and are already orthogonal. We can normalize these … Continue reading

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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

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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

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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

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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

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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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