The general solution for the linear homogeneous system depends only on the freevariables and there is a solution for any choice of these variables. The vectorcoefficients of each one of the free variables are called the fundamental solutions.If coefficient matrix A has dimensions 9x27, is the statement that the linear systemAx = 0 must have at least 18 fundamental solutions correct?TrueO False

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Answered: The general solution for the linear…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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6. Consider the following system of four linear equations in five variables. 2x1 + 4x2 – 2x3 + 2x4 + 4x3 = 2 xị+ 2x2 – x3+ 2x4 = 4 3x1 + 6x2 – 2x3 + x4 + 9x5 = 1 5x1 + 10r2 – 4x3 + 5x4 + 9x5 = 9 | (a) Write a vector equation that is equivalent to this linear system, and a matrix equation that is equivalent to this linear system. (b) Determine the solution set of this linear system, and express it in parametric vector form.
Let Ax = b be a linear system with real coefficients such that A is an m × n matrixand Ax = b has a unique solution. What can you say about m, n? Explain how to picka subsystem of n equations in Ax = b such that the two systems have the same solutions.
1) Find the general solution of the system below. Give your answer as a vector X1 + 2x2 - 3x3 - 0 4x1 + 7x2 - 9x3 = 0 -X1 - 3x2 + 6x3 = 0 A) X1 B) x2 - x3 x2 = x3 3 X3 C) x3 D) X1 x2 - x3 3 x2 X3 X3
Assume that the following augmented matrix represents a system of linear equations. 1 -2 15 18 6. -18 -2 0 -14 0. What kind of solution does this system have? (A) 3 solutions (B) unique solution infinitely many solutions with 4 arbitrary (free) parameters (C) no solution (D) E) infinitely many solutions with 1 arbitrary (free) parameter (F) 2 solutions G) infinitely many solutions with 2 arbitrary (free) parameters (H) infinitely many solutions with 3 arbitrary (free) parameters O A O B OC OD CE OF H.
Select the correct statements from below for a linear system of equations Ax = b. The determinant of a permutation matrix is negative if it leads to an odd number of row interchanges in Gaussian elimination. A zero in the pivot position upon pivoting implies that the matrix A is singular. If A is a lower triangular matrix then the system could be solved by back-substitution. The number of floating point operations in back-substitution is of order O(n³). If the LU decomposition for A is given then the systems could be solved in O(n²) operation.
Is it possible that all solutions of a homogeneous system of twelve linear equations in fourteen variables are multiples of one fixed nonzero solution? Discuss. Consider the system as Ax = 0, where A is a 12x 14 matrix. Choose the correct answer below. O A. Yes. Since A has at most 12 pivot positions, rank As 12. By the Rank Theorem, dim Nul A = 14- rank Az 2. Since there is at least one free variable in the system, all solutions are multiples of one fixed nonzero solution. O B. No. Since A has 12 pivot positions, rank A= 12. By the Rank Theorem, dim Nul A = 12 - rank A= 0. Since Nul A = 0, it is impossible to find a single vector in Nul A that spans Nul A. OC. No. Since A has at most 12 pivot positions, rank As 12. By the Rank Theorem, Nul A= 14 - rank Az 2. Thus, it is impossible to find a single vector in Nul A that spans Nul A. O D. Yes. Since A has 12 pivot positions, rank A= 12. By the Rank Theorem, dim Nul A= 12- rank A= 0. Thus, all solutions are multiples of one fixed nonzero…
Given:2a + 5b + 3c = 23a + b - 2c = 3-a + 2b + c = 1Identify the coefficient matri A and the augmented matrix A'of the given linear system.
4) Consider the system of linear equations: X1 + 5x2 + 3x3 = 4 X1 – X2 + 6x3 = 16 2x1 + x2 = 5 a) Write the system as Ax=b b) Solve the system using Gaussian Elimination with backward substitution. c) What is the determinant of A. d) Perform one step of the Gauss-Seidel Method on this system starting with initial guess 1 1
2 and u = 4 Write as the sum of two orthogonal vectors, one in Span {u} and one orthogonal to u. Let y = - 8 y=ý+z=D+0 (Type an integer or simplified fraction for each matrix element. List the terms in the same order as they appear in the original list.)
1. A system of linear equations can be described in a matrix equation Ax = b with 2 -1 1] A = |1 1 2,x = |y| and b = |b2 [-1 5 4] a. Determine if the columns of the matrix A span R³. Give your arguments. b. If the system is homogeneous, describe the trivial solution of the system (if any) in parametric vector form. Also, give a geometric description of the solution set. Give the clear argument. c. If the system is nonhomogeneous and consistent, describe b in parametric vector form. Give the clear argument. d. If b, = 5, b2 = -3, bz = 0, is the system consistent? Give valid argument. Find the solution set of the system. Compare the solution set with the trivial solution of the homogeneous system in b.
8. Solve the linear system Ax-b by x₁ + x₂-3x₂ = -1 -x₁ + 2x₂ = 1 x₁ - x₂ + x₂ = 2 a) finding the LU-factorization of the coefficient matrix A b) Solving the lower triangular system Ly-b. c) solving the upper triangular system Ux-y.
Describe all solutions of Ax = 0 in parametric vector form, where A is row equivalent to the given matrix. [:::) 1 2 0 - 3 3 6 0 - 9 x=x+x+xO X= X2 + X3 +X4 (Type an integer or fraction for each matrix element.)

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