- June 22, 2020

Semester Two Final Examinations, 2019 ENGG7302 Advanced Computational Techniques in Engineering Page 1 of 3 This exam paper must not be removed from the venue School of Information Technology and Electrical Engineering EXAMINATION Semester Two Final Examinations, 2019 ENGG7302 Advanced Computational Techniques in Engineering This paper is for St Lucia Campus students. Examination Duration: 90 minutes Reading Time: 10 minutes Exam Conditions: This is a Central Examination This is a Closed Book Examination – no materials permitted During reading time – write only on the rough paper provided This examination paper will be released to the Library Materials Permitted In The Exam Venue: (No electronic aids are permitted e.g. laptops, phones) Calculators – Casio FX82 series or UQ approved (labelled) Materials To Be Supplied To Students: 1 x 14-Page Answer Booklet Instructions To Students: Additional exam materials (eg. answer booklets, rough paper) will be provided upon request. Venue ____________________ Seat Number ________ Student Number |__|__|__|__|__|__|__|__| Family Name _____________________ First Name _____________________ For Examiner Use Only Question Mark Total ________ Semester Two Final Examinations, 2019 ENGG7302 Advanced Computational Techniques in Engineering Page 2 of 3 Part A. (30 marks in total, 5 marks each) For each question, select the correct answer (only one option is correct among the four ones; write down your answer in the answer booklets.) 1. Consider the full singular value decomposition (SVD) of a matrix VH, and A , . Consider the following statements, [1] U, V must be orthogonal matrices; [2] ; [3] may have min non-zero singular values; [4] U, V may have the same dimension. Which of the following is correct? (a) [1], [2], [3], [4] (b) Only [1], [2], [3] (c) Only [3],[4] (d) None of [1], [2], [3], [4] 2. Consider a matrix and its third row vector. What is the difference between their -norms? (a) 4 (b) 3 (c) 2 (d) 0 3. Consider the 2-norm and Frobenius norm of a matrix A. Which of the following is correct? (a) Its 2-norm is always larger than its Frobenius norm (b) Its 2-norm is always smaller than its Frobenius norm (c) Its 2-norm can be larger than its Frobenius norm (d) Its 2-norm cannot be larger than its Frobenius norm 4. Consider a matrix A . It may have the following properties, [1] A must have a pseudo-inverse; [2] range(A) ; [3] rank(A) is always equal to m; [4] 0 is not an eigenvalue of A. Then which of the following is always correct? (a) Only [2] (b) Only [1],[3] (c) [1], [2], [3], [4] (d) Only [1],[2] 5. If P is a projection matrix, then it may have the following properties, [1] P3=P2; [2] range[I-P] = null(P), where I is the identity matrix; [3] if P is an orthogonal projector, then it must be a symmetric matrix; [4] it must be a square matrix. Then which of the following is always correct? (a) Only [1],[2],[4] (b) Only [2],[4] (c) [1], [2], [3], [4] (d) Only [3], [4] 6. If A is a unitary matrix, consider the following statements: [1] its singular value decomposition (SVD) is VH, must be an identity matrix; [2] its eigenvalues are equal to one. Which of the following is correct? (a) [1], [2] (b) Only [1] (c) Only [2] (d) Neither [1] nor [2] ´Î!m n £m n ( , )m n ¥ ´Î =,m n m nC Î mC Semester Two Final Examinations, 2019 ENGG7302 Advanced Computational Techniques in Engineering Page 3 of 3 Part B. (70 marks in total) Question 7. (25 marks) Consider a 2×2 matrix A . Perform the following calculations. (a) If – Calculate its full SVD (that is, find its U, , V ); – Show that U is an unitary matrix; – Find the null-space, range, rank and condition number of matrix A; (b) If Use a low rank approximation (using rank r =1), to form a new matrix B, please calculate the Frobenius norm of (A-B). Question 8. (15 marks) Find (a) the projection of vector b on the column space of matrix A and (b) The projection matrix P that projects any vector in (two-element vector) to the column space of matrix A. Note: Question 9. (20 marks) Consider a linear system Ax=b, and the SVD of the matrix VH. – Use matrices U, , V to express the pseudo-inverse of the linear system; – Show that , where are the first column vectors of matrices U, V respectively, and is the largest sigular value. Question 10. (10 marks) Consider the following claim: “There exists a matrix A for which the range and null space are identical.” If you think such a matrix can exist, give an example and show the matrix’s range and null space; otherwise, give your reason to reject this claim. END OF EXAMINATION a b c d é ù = ê ú ë û 1, 0, 0, 2a b c d= – = = = – 1000, 0, 0, 1a b c d= = = = 1 1 é ù = ê ú ë û 1 0 1 1 é ù = ê ú ë û R2 12 1 1 1 1 1 1 2 – é ù é ù ê ú ê ú ë û ë û – = – 1 1 1 1U = AV σ 1 1 U ,V 1σ ´Î 4 4R