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Tag Archives: orthogonalization
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 GramSchmidt orthogonalization. Answer: … Continue reading
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
Linear Algebra and Its Applications, Exercise 3.4.26
Exercise 3.4.26. In the GramSchmidt 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 3.4.24
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 3.4.19
Exercise 3.4.19. When doing GramSchmidt 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
Linear Algebra and Its Applications, Exercise 3.4.16
Exercise 3.4.16. Given the matrix whose columns are the following two vectors and [sic]: factor as . If there are vectors with elements each, what are the dimensions of , , and ? Answer: With and as the two columns … Continue reading
Linear Algebra and Its Applications, Exercise 3.4.15
Exercise 3.4.15. Given the matrix find the orthonormal vectors and that span the column space of . Next find the vector that completes the orthonormal set, and describe the subspace of of which is an element. Finally, for find the … Continue reading
Linear Algebra and Its Applications, Exercise 3.4.14
Exercise 3.4.14. Given the vectors find the corresponding orthonormal vectors , , and . Answer: We first choose . We then have We then have Now that we have calculated the orthogonal vectors , , and , we can normalize … Continue reading