Author Archives: hecker

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

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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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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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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

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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

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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

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Linear Algebra and Its Applications, Exercise 3.4.13

Exercise 3.4.13. Given the vectors and the matrix whose columns are , , and , use Gram-Schmidt orthogonalization to factor . Answer: We first choose . We then have We then have We have , so , , and . … Continue reading

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Linear Algebra and Its Applications, Exercise 3.4.12

Exercise 3.4.12. Given the vectors and , find a scalar such that is orthogonal to . Given the matrix whose columns are and respectively, find matrices and such that is orthogonal and . Answer: We must have . This implies … Continue reading

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Linear Algebra and Its Applications, Exercise 3.4.11

Exercise 3.4.11. If the matrix is both upper triangular and orthogonal, show that must be a diagonal matrix. Answer: Let be an by matrix. Since is upper triangular we have where for . Our goal is to prove that is … Continue reading

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Linear Algebra and Its Applications, Exercise 3.4.10

Exercise 3.4.10. Given the two orthonormal vectors and and an arbitrary vector , what linear combination of and is the least distance from ? Show that the difference between and that combination (i.e., the error vector) is orthogonal to both … Continue reading

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