A=-ΤΟ 1000 0 2000030 00(3)Consider the following linear system: Axb, where A is given by (3), as in Problem 4, and vectorb = [0, 0, 0, 2023]T. Since matrix A is singular, there are infinitely many least squares solutions. Find theleast squares solution x* with the minimal Euclidean norm.

The given matrix is and .We need to find the least square solution of the system with minimal…

Answered: A = 0 1 00 0020 00 03 000 0 = (3)…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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5.(10 pts) Given that = is the exact solution of the linear system of equations Ai = 6 with -2 1 A = 2 -3 (i) Let ze be an approximate solution. Calculate the relative forward error, relative back- ward error, and error magnification factor for this approximate solution (use infinity norm). (ii) Find the condition number of matrix A (using infinity norm). (iii) For a system of linear equations A = 6, what is the possible consequence of an extremely large condition number of A?
The coefficient matrix is not strictly diagonally dominant, nor can the equations be rearranged to make it so. However, both the Jacobi and the Gauss-Seidel method converge anyway. Demonstrate that this is true of the Gauss-Seidel method, starting with the zero vector as the initial approximation and obtaining a solution that is accurate to within 0.01.
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2. If the equation Gx = y has more than one solution for some y & R", can the columns of G span R"? (you can use Invertible Matrix Theorem) Note. If the equation Hx = c is inconsistent for some c = R", what can you say about the equation? (you do not need to submit).
Solve the following system using the Gaussian algorithm: 2x1 – x2 + 3x3 = 1 -2x1 + x2 – X3 = 2 X1 — Зх2 + 6х3 = 3 X1 Let x be the vector x = x,, where x1, x2 and x3 are solutions of the system, and y be the vector y = 1|. Evaluate the (X3) scalar product of x and y. [Please enter your answer numerically in decimal format. You will be marked correct as long as what you enter is within 0.25 of the correct answer. So for example, if the correct answer is 6.78 then any input that lies between between 6.53 and 7.03 will be marked as correct.]
1. In each part of this question, you have been given a system of equations and the corresponding matrix taken to RREF. Write the general solution as a vector equation. • List the basic solutions. a. b. (x₁ - x₂ + 2x4 = 0 [1 -X₁ + x₂ + 3x3 - 11x4 0 2x₁ - 2x₂ + 4x4 0 0 -4x₁ + 4x₂ + 3x3 - 17x4 = 0 Lo - = 0 " -1 0 2 01 0 1 -3 0 0 0 0 0 0 00 (3w+9y15z = 0 1 0 -14w + 33x - 75y - 62z = 0 0 1 -5w+ 14x - 29y - 31z = 0 0 0 (52w 164x + 320y + 396z = 0 Lo 0 3 -1 -4 0 0
Consider the following linear system: 1 2 X1 3 (6) X2 Assume that, due to numerical error, the linear system (6) is solved with the same matrix, but with a slightly different right hand side: 1 2 X1 3.01 (7) 3 4 X2 7.02 Without calculating the solutions of the linear systems (6) and (7), mate the relative error in the 1-norm, | : 1. esti-
2. Consider the function y(x) = 5x over the interval [0, 10]. (a) If the points A(0, y(0)), B(5, y(5)), and C(10, y(10)) are on the curve represented by the above function, find the slope of the lines (secants) AB, and AC, respectively. (b) If x represents time and y represents the displacement of a particle moving along a line, what is the physical meaning of your results to part (a)? (c) Now consider a point D(10 – h, y(10 – h)) on the curve represented by the given function, find the slope of the line DC. (d) As h → 0, what is the limiting value of your result to part (c)? (e) What is its physical meaning of your result to part (d) if again x represents time and y represents the displacement?
5x1 + 6x2 + 11x3 + 7x4 6x5 5 4. Membership in a Span: For every item: (i) apply the Modified Gauss-Jordan Algorith to determine if b is in Span(S); (ii) write the solution vector i if there are solutions to ti corresponding augmented matrix; (iii) express S in the simplest possible way, and check directly that your answer is correct. as a linear combination of the vectors а. Б - (-4, 2, -3); S - {{7,4, -6), (-5, -2, 3)} b. b = (-4, 2,–4); S = {(7,4,–6), (-5,–2, 3)} %3| b = (9, 7,-8, 2); S = {(5,-3, 2, 6), (-2,–3, 5, 8), (–5, 4,–2,–3)} d. b = (-10, 13,–4, 9); S = {(5,-3, 2, 6), (-5, 4,–2,–3), (-5, 7,–2, 6)} Б - (13, 14,-18,-11); S = {(5,-3, 2, 6), (–2,–3, 5, 8), (–5, 4,–2,–3), (11, 10,–13,-6)} b = (8,–9,–8, 15,-3), S = {(6,0,4, 3, 2), (3, 2, 7, 1,–2), (2, 1, 2, –1, 3)} b = (-4,–1, 4, 7,-9), S = {(3, 2, 7, 1,–2), (2, 1, 2,–1, 1), (1, 0,–3,–3, 4)} h. Б - (-3,6, -1,-9, 4), S = {(6,0,–1, 3, 2), (3, 2,–3, 1,–2), (0,–4, 5, 1, 6), (3,–2, 1, 3, 3)} с. %3D е. f. g.
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1. 0 2 4 1 -1 10 0 0 1 1 The reduced row echelon form of the augmented matrix of a linear system is given above. Describe the solution set of this linear system in parametric vector form.

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A = 0 1 00 0020 00 03 000 0 = (3) Consider the following linear system: Ax b, where A is given by (3), as in Problem 4, and vector 6 = [0, 0, 0, 2023] T. Since matrix A is singular, there are infinitely many least squares solutions. Find the east squares solution x* with the minimal Euclidean norm.