Assignment #9: Orthogonal Projections, Gram-Schmidt, and Least Squares. Name:

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1 Assignment 9: Orthogonal Projections, Gram-Schmidt, and Least Squares Due date: Friday, April 0, 08 (:pm) Name: Section Number

2 Assignment 9: Orthogonal Projections, Gram-Schmidt, and Least Squares Due date: Friday, April 0, 08 (:pm) For full credit you must show all of your work.. Consider v, v which form an orthogonal basis for  4, v, v 4 a. ( points) Find the orthogonal projection of onto each of the D subspaces of  4 spanned by each of the basis vectors v i b. ( points) Epress as a linear combination of {v, v, v, v 4 }. c. ( points) Find the closest point ˆ to in the subspace W of  4 spanned by {v, v } d. ( points) Epress as the sum of two orthogonal vectors, w which is in the D subspace W spanned by {v, v } and u which is in the D subspace 8. Let v be a basis for a D subspace of  0, v 0 and let V be the matri 4 6 whose columns are defined by v and v. a. ( points) Does V have orthogonal columns? Is V an orthogonal matri? b. ( points) Compute A VV T and B V T V c. ( points) Let U be a matri with orthonormal columns u and u that span the same subspace of  as is spanned by the columns of V. Is U an orthogonal matri? d. ( points) Compute C UU T and D U T U. How are these results similar to and different from the results A and B from b) above? e. ( points) Compute a Ay and c Cy where y. What do you notice about a and c? W

3 f. ( points) Epress c as a linear combination of u and u. Is it possible to epress y as a linear combination of u and u? How do you know? g. ( points) Epress y as the sum of two orthogonal vectors, one of which is in the subspace spanned by u and u.. For each of the following, choose the correct option. If the answer is True, eplain how you know. If the answer is False, give a counter-eample. (9 points) a. True or False: every matri that has orthogonal columns is an orthogonal matri b. True or False: every linearly independent set of vectors in  is an orthogonal set c. True or False: if two vectors v and v are orthogonal, then they are also linearly independent d. True or False: if two vectors v and v are orthogonal, and neither v nor v equals zero, then v and v are linearly independent e. True or False: if A is an orthogonal matri, then A is invertible f. True or False: if A is an orthogonal matri, then AA T I g. True or False: if A is a matri that has orthogonal columns, then A T A I h. True or False: if U has orthogonal columns, then U T U D where D is diagonal. i. True or False: if A and B are each orthogonal matrices, then their product AB will also be an orthogonal matri 4. (8 points) a. Use the Gram-Schmidt Process to find an orthogonal basis for the subspace of  4 spanned by: v, v, v b. Use the Gram-Schmidt Process to find an orthogonal matri Q whose first three columns form an orthonormal basis for the subspace of  4 spanned by v, v, v above.. ( points) What complications will arise when you try to use the Gram-Schmidt process to find an orthogonal basis for the column space of an m n matri whose columns are linearly dependent? Use a robust variant of the Gram-Schmidt process to find an orthogonal basis for the column space of: A, B, C

4 6. (8 points) Using Matlab, write a function GramSchmidt() that takes as input an arbitrary m n matri M and produces as output an m p matri B whose columns form an orthogonal basis for the subspace of  m spanned by the columns of M. You do not need to normalize the columns of B. Test your program on the matrices in question above to verify that your program produces valid results under challenging circumstances. Please pass in a printout of your code and its output (6 points) Working by hand, find the QR decomposition of A, where Q is a matri whose columns form an orthonormal basis for the column space of A and R Q T A is a invertible, upper triangular matri such that A QR. Hint: the columns of Q can be obtained from the columns of A using Gram-Schmidt orthogonalization (6 points) Working by hand, find the QR decomposition of A, where Q is a orthogonal matri and R Q T A is a upper triangular matri such that A QR. 9. (6 points) Find the Householder reflection matri Q that will rotate the vector into alignment with e (7 points) Use Householder reflections to derive the QR decomposition of A 6 0. (6 points) Find the best-fitting line (in the least-squares sense) to these four points: (, ), (0, ) (, 4), (, 4). Use Matlab or an equivalent graphing program to visualize your results, plotting both the original points and the fitted line. What is the least squares error in this approimation?. (6 points) Find the best-fitting quadratic (in the least-squares sense) to these four points: (, ), (, ), (, 0), (, ). Use Matlab or an equivalent graphing program to visualize your results, plotting both the original points and the fitted curve. For full credit you must show all of your work. You can check your answers using the Matlab function polyfit().. (6 points) Find the best-fitting plane (in the least-squares sense) to these four points: (,, ), (, 0, ), (0,, ), (,, ). For full credit you must show all of your work. You can check your answer using Matlab: coeffs [ y c]\z where,y,z are vectors containing the, y, z coordinates of the input data points and c is a vector of the same length containing all s.

5 4. ( points) Which of the systems listed below has a unique least squares solution? Eplain how you can tell without calculating the least squares solution