1 32 -1 02 1 -21 6116.1 -1A =-31132. Consider the matrix A in Problem 1.(a) Find the solution space of the homogeneous system Ax = 0, that is N(A), thenullspace of A.(b) Find the basis and dimension of N(A).1-1If b =determine whether the nonhomogeneous system Ax = b is consis-21tent.(c). If the system Ax = b is consistent where b is given in 2 (b) find the completesolution in the formx = Xp + Xhwhere x, denotes a particular solution and x, denotes a solution of the associatedhomogeneous system Ax = 0.

Null space of the matrix is obtained by solving the system of linear equation Ax=0 To obtained the…

1 1 3 2 -1 0 1 1 A = -3 2 1 -2 1 6 1 4 1 3 2. Consider the matrix A in Problem 1. (a) Find the solution space of the homogeneous system Ax = 0, that is N(A), the nullspace of A. (b) Find the basis and dimension of N(A). 1 If b = -1 determine whether the nonhomogeneous system Ax = b is consis- 1 tent. (c), If the system Ax = b is consistent where b is given in 2 (b) find the complete solution in the form x = Xp + Xh where x, denotes a particular solution and x, denotes a solution of the associated homogeneous system Ax = 0.

1 1 3 2 -1 0 1 1 A = -3 2 1 -2 1 6 1 4 1 3 2. Consider the matrix A in Problem 1. (a) Find the solution space of the homogeneous system Ax = 0, that is N(A), the nullspace of A. (b) Find the basis and dimension of N(A). 1 If b = -1 determine whether the nonhomogeneous system Ax = b is consis- 1 tent. (c), If the system Ax = b is consistent where b is given in 2 (b) find the complete solution in the form x = Xp + Xh where x, denotes a particular solution and x, denotes a solution of the associated homogeneous system Ax = 0.

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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Question

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1 3
2 -1 0
2 1 -2
1 6
1
1
6.
1 -1
A =
-3
1
1
3
2. Consider the matrix A in Problem 1.
(a) Find the solution space of the homogeneous system Ax = 0, that is N(A), the
nullspace of A.
(b) Find the basis and dimension of N(A).
1
-1
If b =
determine whether the nonhomogeneous system Ax = b is consis-
2
1
tent.
(c). If the system Ax = b is consistent where b is given in 2 (b) find the complete
solution in the form
x = Xp + Xh
where x, denotes a particular solution and x, denotes a solution of the associated
homogeneous system Ax = 0.

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2. Consider the following system of linear equations: 20 (20 21) (2) - () We know the inverse of the coefficient matrix, denoted by A, is (-21 20 20 -19, (a) What is the condition number of A (that is, ||A||||¹||)? (b) Is the system well-conditioned with respect to perturbations of the right-hand side
Theorem 1. The following system of equations Un-1Vn-3 Un-3Vn-1 Un+1 = and vn+1 = (35) Vn-1(A+ Bun-1Vn–3) Un-1(C+ Dun-3Vn–1)' (1-А) has a 2-periodic solution if u_3 = u_1, v_3 = v_1, u_3v_3 = u_2v_2 = " and A = 42 C, B = D+ 0. 41 -A), where A = C,B = Proof. Let u _3 = u_1,v_3 = v_1 and u_3v_3 = u_2v_2 = D in the exact equation (34a). Then 2m-1 2m + (1-C) Σ C n-1 l=0 U4n-2 =u-2 |T 2m A2m+1 + (1– A) E A! m=0 1=0 =u-2. (36)
2. Consider the following are augmented matrices. (a) 1 0 2 4 0 1 -3 5 -6 0 0 0 0 00 (b) 2 4 1 1 0 0 1-3 5 -6 0 0 10 0 0 1 (c) 1 -2 0 3 0 4 1 2 0 9 0 0 18/ b Enimonb lo olnn 1, 0 0 पतं • List the free variables. • Write the general solution of the corresponding system of linear equations. • What is the rank of the coefficient matrix? What is the rank of the augmented matrix?
If A is a constant matrix, the fundamental matrix of the system x'(t) = Ax(t) is non-constant. O True False
2. (a) Use the linearity of matrix-vector multiplication to prove the following: if the vectors x1, X2 are solutions to Ах = 0 (3.1) then the linear combination x = cịx1 + c2x2 is also a solution, for any constants c1, c2. (You may use the results you showed in the previous problem) (b) Show that the results is not true for nonhomogeneous systems: if the vectors x1, X2 are solutions to Ax = b (3.2) with b + 0, then the linear combination x = c1x1 + c2x2 is not a solution. (c) Show that if, however, x1 and x2 are solutions to the homogenous problem (3.1) and is a particular solution to the nonhomogeneous problem (3.2), then Хр х— С1X1 + C2X2 + Хр is a solution for the nonhomogeneous problem (3.2), for any c1, C2.
2. please solve it on paper