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

Problem 1RQ

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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…
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Hello, I am not confident in this Linear Algebra question: Suppose that a nonhomogeneous system with 10 linear equations in 8 unknowns has a solution with 2 free variables. Is it possible to change some constants on the equations’ right hand side to make the new system inconsistent? Explain in detail. Help would be greatly appreciated
It is a basis for the S Solution space of the system
Let A and B be matrices of thesame size. If x is a solution to both the system Ax =0 and the system Bx = 0, show that x is a solution tothe system (A + B)x = 0. Provide reasoning/details please.
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.

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