ACS I - Problem Set # 1 Problems # 2 ,4,5,8 are to be turned in Due: Thursday, 1/22 1. a. First determine if each of the sets is a vector space when the standard operations of addition and scalar multiplication are used. If the space is finite dimensional determine a basis and give its dimension; otherwise indicate that the space is infinite dimensional. i. All 3 × 3 upper triangular matrices. ii. All 3 × 3 unit lower triangular matrices. b. Does each set form a subspace of the give space? i. All points in IR2 which lie on the line y − x = 1. ii. All points in IR2 which lie on the line y − x = 0. iii. All 2x2 diagonally dominant matrices. (The underlying space is all 2x2 matrices which is a vector space.) 2. Analyze the four fundamental spaces for the matrix 0 2 1 4 4 1 A = @ −4 2 −2 0 A 0 4 6 2 by finding the dimension of each space and a basis for each space. Indicate whether each space is a subspace of IR3 or IR4. Give the rank of A. Verify the orthogonality of the appropriate spaces. 3. Let Mk be a Gauss transformation matrix which zeros out entries in the kth column of a matrix below the diagonal. a. Determine M2 where 0 1 2 3 4 1 0 2 5 6 A = B C : @ 0 8 1 0 A 0 10 0 1 Give the inverse of M2. b. Let M1 and M2 be generic 3 × 3 Gauss transformation matrices. Compute the product M2M1. Is the result a Gauss transformation matrix? If not, what properties does it have? Compute the product (M2)−1(M1)−1. Is the result a Gauss transformation matrix? If not, what properties does it have? 4. Consider the variant of LU given by A = LDU where L is a unit lower triangular matrix, D is a diagonal matrix and U is a unit upper triangular matrix. a. If you are given this decomposition, demonstrate how it could be used to solve the linear system A~x = ~b. b. Assume that A is an n × n tridiagonal matrix. Determine the equations for the entries of L, D and U in terms of entries aij and previously calculated `ij, di or uij. Then write your equations in an storage efficient algorithmic format using a for loop; i.e., indicate whether you can overwrite entries of L, D or U on A. c. Determine the operation count for determining L. d. Write a computer code to implement your algorithm in (b). Test your code on the n × n linear system where 0 1 4 −1 0 ··· 0 0 1 0 1 3 B 2 4 −1 0 ··· 0 C 1 B C B 5 C B 0 2 4 −1 0 C B 1 C B . C A = B . C ~x = B . C ~b = B . C B .. .. .. C @ . A B C B C @ 5 A @ 0 2 4 −1 A 1 6 0 2 4 1 for n = 100. Output the solution for the first ten entries of ~x, the `2 and`1 error norms. 5. Use GE to solve the system A~x = ~b by hand using (i) no pivoting (ii) partial pivoting, and (iii) full β pivoting. Use only three digit arithmetic, i.e., numbers represented as 0:d1d2d3 × 10 . Discuss and compare your results. 0:001 1:00 1:00 A = ~b = 1:00 2:00 3:00 The exact solution is ~x = (1:002; 0:998)T and to three digits (1:00; 1:00)T . 6. Symmetric, positive definite (spd) matrices a. Let B = AT A. Is B symmetric? Prove or disprove. Is B positive definite? If not, under what conditions on A is it positive definite? b. If A is an n × n spd matrix then aii > 0 for all i = 1; : : : ; n. Prove or disprove this statement. c. If A is an n × n symmetric matrix where aii > 0 for all i = 1; : : : ; n, then A is a spd matrix. Prove or disprove this statement. d. If A is an n × n symmetric matrix and A = LLT where L is a lower triangular matrix with positive diagonal entries, then A is a spd matrix. Prove or disprove this statement. 7. Matrix norms a. Calculate kAk1, kAk1 and kAkF (the Frobenius norm) where 0 1 −5 8 1 A = @ 2 4 0 A 3 −11 4 b. Demonstrate that kAk1 for an n × n matrix A can be computed by taking the maximum column sum. c. Let A be an general invertible 2 × 2 matrix with entries a b A = : c d Calculate a general formula for K1(A) and comment on when the condition number will be large. 8. If we have A−1 \in hand" then solving the linear system A~x = ~b can be done by multiplying ~b by A−1 which requires O(n2) operations. However, this approach is never recommended computationally. a. Devise an efficient direct algorithm to calculate A−1. b. Determine the operation count for solving A~x = ~b using your algorithm for A−1 from (a). Compare your operation count with solving a linear system by LU factorization. You may assume that LU factorization or GE requires O(n3) operations. 2.