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Author Archives: hecker
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
Linear Algebra and Its Applications, Exercise 3.4.13
Exercise 3.4.13. Given the vectors and the matrix whose columns are , , and , use GramSchmidt orthogonalization to factor . Answer: We first choose . We then have We then have We have , so , , and . … Continue reading
Posted in linear algebra
Tagged orthogonal matrices, orthogonalization, orthonormal vectors
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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
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
Posted in linear algebra
Tagged diagonal matrix, orthogonal matrices, upper triangular matrix
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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
Linear Algebra and Its Applications, Exercise 3.4.9
Exercise 3.4.9. Given the three orthonormal vectors , , and , what linear combination of and is the least distance from ? Answer: Any linear combination of and is in the plane formed by and . The combination closest to … Continue reading
Linear Algebra and Its Applications, Exercise 3.4.8
Exercise 3.4.8. Project the vector onto the two nonorthogonal vectors and and show that the sum of the two projections does not equal (as it would if and were orthogonal). Answer: The projection of onto is . We have and … Continue reading
Linear Algebra and Its Applications, Exercise 3.4.7
Exercise 3.4.7. Given where are orthonormal vectors, compute and show that Answer:We have so that since the transpose of a sum is equal to the sum of the transposes. The product of the sums can then be decomposed into two … Continue reading