Problem 20. Farkas' theorem tells us the following. Let A be an mxn matrixover R and c be an n-vector in R". Then exactly one of the following systemshas a solution:System 1: Ax < O and c™x > 0 for some x E R".System 2: A"y = c and y > 0 for some y e R™.Let1 0 10 1 01 0 1A =c =11Find out whether system (1) or system (2) has a solution.

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Answered: Problem 20. Farkas' theorem tells us…

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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7. Let A be an invertible n x n matrix over R. Consider the system of linear equation Ax = b or 2aija; = bị, AijX'j i = 1,..., n. Let A = C - R. This is called a splitting of the matrix A and R is the defect matrix of the splitting. Consider the iteration Cx(k+1) = Rx(k) +b, k = 0, 1,2,.. . Let (3) :). 4 0 0 4 0 0 0 4 (3) 4 -1 -1 4 -2 A = -1 C = b = 2 *(0) 4 2 The iteration converges if p(C-!R) < 1, where p(C-1R) denotes the spectral radius of C-1R. Show that p(C-!R)< 1. Perform the iteration.
part E and F urgntly need
Let A be a 2 x 2 diagonalizable matrix. 3et+ ecit If is the solution to the system ở' (t) = Ax(t), 3e²t + ec₂t then check All the possible values of c₁ and c2 below. (a)c₁ = 1; (b) C₁ = 2; (c) C₁ = 3; (d) c₂ = = 1; (e) c₂ = 2; (f) C₂ (a) (b) (c) (d) (e) (f) U 1 U U = 3.
1) Please help with following multiple choice ASAP!
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.
Verify that X(t) is a fundamental matrix for the given system and compute X1(t). Then use the result that if X(t) is a fundamental matrix for the system x' = Ax, then x(t) = x(t)X(0)x is the solution to the initial value problem x' = Ax, x(0) = x0 - 24e-1 -9e-21 060 -5 x'= 10 1x, x(0)= 8e3t x(t)= 1 1 0 -4e-t 3e-21 3t 4e -2 -20e-t 3e-21 3t 4e 060 (a) If x(t) = [×₁ (t) x2(t) ×3(t)] and A = 10 1 validate the following identities and write the column vector that equals each side of the equation. 1 1 0 ×₁' = Ax₁ = x2' = Ax₂ = x3' = Ax3 (b) Next, compute the Wronskian of X(t). W[x()()()]= Since the Wronskian is never and each column of X(t) is a solution to x' = Ax, x(t) is a fundamental matrix. (c) Find x(t) = ☐ (d) x(t) = x(t)x¯1 (0)x₁ =
2. A is a 4 x 5 matrix with four pivot positions. a) Does the equation Ax = 0 have a nontrivial solution? b) Does the equation Ax = b have at least one solution for every possible b?
2. Show that if the rank of the m-by-n constraint matrix A is m, then (AM¯'A" )* = (AM¯'A" )' -1 -1 T-1 where M > 0 is the mass matrix.
2. Imagine that, instead of estimating y = XB + ɛ, you estimate y = Zx + ɛ where Z = XP with P being a nonsingular matrix that transforms the columns of X. For example, you can imagine that P would change the measurement units of some variables in X (e.g. going from cm to meters). a) Show that the OLS estimator of y will equal P-'b where b is the OLS estimator of ß. b) Show that the R-squared of the two regressions will be identical.
Please solve as soon as possible
If A is m x n matrix and Ax = b is solvable for any b, then O a. C(A) = R" O b. C(A) is a plane in R" O c. C(A) is a plane in Rm O d. C(A) = RT

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