-
Archives
- October 2021
- January 2021
- March 2019
- January 2018
- December 2017
- January 2017
- December 2016
- November 2016
- October 2016
- September 2016
- July 2016
- October 2014
- June 2014
- May 2014
- April 2014
- March 2014
- February 2014
- January 2014
- December 2013
- November 2013
- October 2013
- September 2013
- August 2013
- July 2013
- June 2013
- May 2013
- April 2013
- March 2013
- February 2013
- January 2013
- November 2012
- October 2012
- September 2012
- August 2012
- July 2012
- June 2012
- May 2012
- April 2012
- September 2011
- August 2011
- July 2011
- June 2011
- May 2011
- April 2011
- March 2011
- January 2011
- August 2010
- June 2010
- May 2010
- November 2009
-
Meta
Monthly Archives: December 2017
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
Posted in linear algebra
Tagged orthogonalization, orthonormal vectors, projection matrix
Leave a comment
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
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
Posted in linear algebra
Tagged orthogonal matrices, orthogonalization, orthonormal vectors
Leave a comment
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
Leave a comment
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
My Math Homework 