CAAM 310 Supplement Problems ================================ The list consists of supplement problems for homework assignments. The actual assignments will be specified by the instructor. Problem 1. Use Gauss-Jordan elimination (Section 2.4) to find the inverse of [ 1 1 1 ] A = [ 0 1 1 ]. [ 0 0 1 ] Problem 2. Use Gauss-Jordan elimination (Section 2.4) to find the inverse of [ 1 3 0 ] A = [ 2 5 1 ]. [ -3 -9 -1 ] For Problems 3, 4 and 5: Give an example to prove 'yes' or a reason to prove 'no'. A unit lower triangular matrix L is a lower triangular matrix with ones on the diagonal. For the following questions, can you find a unit lower triangula matrix L such that: Problem 3. [ 1 3 0 ] [ 1 3 0 ] L [ 2 5 1 ] = [ 3 8 1 ] ? [ -3 -9 -1 ] [ 0 1 0 ] Problem 4. [ 1 -5 0 ] [ 1 -5 0 ] L [ 0 1 0 ] = [ 2 -9 0 ] ? [ 0 0 0 ] [ 1 -2 0 ] Problem 5. [ -1 3 -2 ] [ -3 3 2 ] L [ 2 -6 2 ] = [ 1 2 1 ] ? [ 0 0 0 ] [ -2 5 3 ] Problem 6. (a) Find the symmetric factorization A = L D L^{T} for [ 2 4 4 ] A = [ 4 7 10 ]. [ 4 10 7 ] (b) Find L^{-1} by Gauss-Jordan elimination. Problem 7. Find the symmetric factorization L D L^{T} for A^{-1}, where the matrix A is given in Problem 6a. (If helpful, re-use any results for L, D, L^{-1} from Problem 6.) Problem 8. Find the quadratic polynomial p(x) = a + bx + cx^{2} that fits the points: (1, 4), (2, 7), (3, 16). Problem 9. Find the quadratic polynomial p(x) = a + bx + cx^{2} that fits the points: (-2, 3), (0, -1), (1, 0). Problem 10. If possible, express the polynomial p(x) = 5 - 6x + x^2 as a linear combination of the polynomials: 1, 2 + x, 3 + 2x + x^2. Problem 11. If possible, express the polynomial p(x) = -2 + 2x + 4x^2 as a linear combination of the polynomials: 1 + x, 2 + x^2, 3 - x + 2x^2. For Problems 12 and 13: A company uses three oil refineries. Each refinery produces three petroleum-based products: heating oil, diesel oil, and gasoline. From one barrel of petroleum, Refinery 1 produces 10 gallons of heating oil, 30 gallons of diesel oil, and 50 gallons of gasoline. If one barrel of petroleum is used, the output from the other refineries are also included below. Refinery 1 Refinery 2 Refinery 3 Heating Oil 10 20 10 Diesel Oil 30 70 20 Gasoline 50 100 60 Problem 12. Suppose that we have a daily demand for 240 gallons of heating oil, 670 gallons of diesel oil, and 1280 gallons of gasoline. How many barrels of petroleum need to be used at each refinery so that the total output of the 3 refineries satisfies the daily demand? Problem 13. Suppose that we have a daily demand for 200 gallons of heating oil, 600 gallons of diesel oil, and 1050 gallons of gasoline. How many barrels of petroleum need to be used at each refinery so that the total output of the 3 refineries satisfies the daily demand? Matlab can be used for Problems 14 - 16: Problem 14. [ -14 0 ] If possible, express the 2 by 2 matrix E = [ -28 64 ] as a linear combination of the matrices [ 3 1 ] [ 13 3 ] [ -4 0 ] [6 1 ] M1 = [ -5 -4 ], M2 = [ -7 -28 ], M3 = [ -10 0 ], M4 = [3 4 ]. Problem 15. [ -5 1 ] If possible, express the 2 by 2 matrix E = [ -4 -4 ] as a linear combination of the matrices [ -4 -4 ] [ -3 -1 ] [ -2 2 ] [ -1 5 ] M1 = [ -4 -4 ], M2 = [ -1 -1 ], M3 = [ -3 -3 ], M4 = [ 10 13 ]. Cut-and-paste the following matrices into MATLAB when answering the questions below. A = [-2 4 6 10 1 -10; 4 -8 2 1 12 -8; 2 -4 2 2 7 -6] B = [-1 1 -3 -2; 2 -2 6 4; 1 -1 3 2] Problem 16. (a) Give a basis for the column space of A. (b) Give a basis for the column space of B. Fill in the blank: (c) The column space of A spans a _(blank)_ dimensional subspace in R^3. (d) The column space of B spans a _(blank)_ dimensional subspace in R^3. Problem 17. (a) Give a vector b1 that lies in the column space of A, and that also lies in the column space of B. (b) Give a vector b2 that lies in the column space of A, but does not lie in the column space of B. (c) Give a vector b3 that does not lie in the column space of A, and does not lie in the column space of B. Problem 18. Give a particular solution to Ax = b, where b = [7; 7; 5] Note: a "particular" solution expresses b as a linear combination of the pivot columns of A. Problem 19. Use MATLAB to help you find the complete solution to Bx = b, where b = [-5; 10; 5] For Problems 20 - 23: (1) Answer the 4 questions below. Note that each one has multiple parts. (2) You can do the set by entirely hand, entirely in MATLAB, or any linear combination of these methods. Be neat. (3) A^T means the transpose of A. In MATLAB, A' gives the transpose of A. Problem 20. (a) Determine the complete solution to Mx = b, where M = [ 2 -1 2 7 21 14; -1 2 -2 -7 -21 -14; 2 -2 5 14 42 28 ]; and b = [5; 0; 8]; (b) Determine the complete solution to Nx = d, where N = [ 1 -1 2 2 3 4; 3 -3 6 3 10 12; 2 -2 4 2 6 10 ]; and d = [6; 14; 12]; (c) Find a solution y that solves both My = b and Ny = d. Problem 21. The matrix A is: A = [ 1 -1 -1 1 1; 3 -2 -1 4 5; 1 2 5 4 7 ]; (a) Find the particular solution to Ax = b, where b = [-5; 0; 40]; (b) Give a 5 by 5 matrix B with the following properties: The first r columns of B are a basis for the row space of A, and the last (5-r) columns of B are a basis for the nullspace of A. Note: r is the rank of A. (c) From 2(b), find a solution to Bc = [6; -5; -4; 7; 8]; The solution c indicates that [6; -5; -4; 7; 8] is in which subspace of A? (d) From 2(b), find a solution to Bc = [-15; -12; 1; -2; 6]; The solution c indicates that [-15; -12; 1; -2; 6] is in which subspace of A? (e) From 2(b), find a solution to Bc = [10; 15; 0; 0; 0]; (f) Use your result from 2(e) to help answer the following: Suppose [10; 15; 0; 0; 0] = v + w where v is in the row space of A and w is in the nullspace of A. Find v. Find w. Compute A * v. Compute A * w. Problem 22. A = [ 1 1 1 0; 1 1 -1 0; 1 -1 0 1 ]; (a) Give a basis for the nullspace of A^T * A. (b) Give a basis for the nullspace of A. (c) Can you find a vector x such that (A^T * A) x equals the zero vector, but Ax does not equal the zero vector? If 'yes', give the vector x. If 'no', briefly explain why it is not possible. Problem 23. A = [ 1 1; 1 1; 1 -1 ]; (a) Give a basis for the column space of A * A^T. (b) Give a basis for the column space of A. (c) Can you find a vector b such that (A * A^T) x = b is solvable, but Ax = b is not solvable? If 'yes', give the vector b. If 'no', briefly explain why it is not possible. Problem 24. Let a = [1;0;1], b = [1;0;0], c = [2;1;0]. (a) Use the Gram-Schmidt process to produce a set of orthonormal vectors q1, q2, q3. (b) Give a QR factorization of A, where the columns of A are the vectors a, b, c. Problem 25. Let A = [1 1; 0 1; 0 1]. (a) Compute the projection matrix P = A*inv(A'*A)*A'. (b) Find an orthonormal basis for the left nullspace of A. Find an orthonormal basis for the left nullspace of P. In R^3, do the columns of A span the same subspace as the columns of P? (c) If a vector b lies in the left nullspace of A, what is the least squares solution to Ax = b? Explain. Problem 26. For every matrix A, prove that A and AA^{T} have the same column space. For Problems 27 - 29: After you explain how you constructed the answer to each problem below, type it into MATLAB and generate some output that demonstrates its correctness. You may use MATLAB to help construct the answer. In demonstrating your answer, you will need to use MATLAB commands such as: inv(A) ~ computes the inverse of A A^50 ~ raises A to the power 50 rank(A) ~ computes the rank of A sum(A) ~ computes the sum of each column of A. Problem 27. Find a 2 by 2 matrix A such that as k -> infinity, [ 1/2 1/2 ] lim A^(-k) -> [ ]. [ 1/2 1/2 ] * Note that we are taking the powers of the inverse of A. * Compute powers for k = 2, 10, 50 to demonstrate convergence. Problem 28. Find a 4 by 4 positive Markov matrix A1 of rank 1. Give other 4 by 4 positive Markov matrices A2, A3, A4 with rank 2, rank 3, and rank 4, respectively. * Use rank(A1) and sum(A1) to show that A1 is a rank 1, Markov matrix. * Do the same demonstration for A2, A3 and A4. Problem 29. Find a matrix A such that as k -> infinity, [ 1/3 1/3 1/3 ] lim A^k -> [ 1/3 1/3 1/3 ]. [ 1/3 1/3 1/3 ] Do not use the limit matrix (above) as your initial matrix A. * Compute A^k for k = 2, 10, 50 to show that it is converging * to a matrix whose entries are all 1/3. Please do the following problems by hand. Exception: You can use MATLAB to help you solve Problem 27. Note that i is the imaginary number sqrt(-1). Problem 30. The following information is known about the matrix A: [ 1 ] [ 8 ] [ 1 ] [ 14 ] [ 1 ] [ 18 ] A [ 1 ] = [ 2 ], A [ 2 ] = [ -4 ], and A [ 3 ] = [ -6 ]. [ 0 ] [ -2 ] [ 1 ] [ -2 ] [ 1 ] [ -6 ] Find the matrix A. Problem 31. Give the A = LU factorization of [ 1 -i (2+i) ] A = [ ]. [ i (1+i) (-4+2i) ] Problem 32. Find a nonzero vector w that is orthogonal to the vectors [ 1 ] [ i ] u = [ i ] and v = [ 0 ]. [ 1+i ] [ 0 ] Problem 33. Find the inverse of the matrix [ 1 (2+i) ] A = [ ]. [ (2-i) -1 ] Is the inverse a symmetric matrix? Explain. Is the inverse a Hermitian matrix? Explain.