2.5. Consider the linear programminimizesubject toz = c'xAx ≤ be'x = 1X₁X120, x,, free,where e = (1..... 1). b and c are arbitrary vectors of length n. and A is the matrixwith entries a₁ = an= = 1 for i = 1,...,n and all other entries zero. Use theconstraint ex = 1 to eliminate the free variable x,, from the linear program (as inthe previous problem). Is this a good approach when n iş large?

Given: Consider the linear program minimize z=cTxsubject to Ax≤b…

Answered: Consider the linear program minimize…

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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31. Construct a nonzero 2 x 2 matrix that is invertible but not diagonalizable.
LOOK AT IMAGE Write down the dual of the above LP (for general n and using the notation provided, not for anexample). Do not leave your LP in matrix form, specify the constraints.2. Write the complementary slackness conditions for the above LP and its dual.
10. Let the null space of matrix G and the null space of matrix H be orthogonal complements. Then only one of the following statement can be true: a. N(G) = N(H) b. N(G")=N(#") © N(G)=C(H") d. N(G)=C(H)
Let A be a matrix and mathbfb be a matrix and a vector defined by 1 2 1 -1 1 b = 1 1 3 Compute x* that minimizes ||Ax - b|.
I have ans but I need explation Solution: The rows of a Jacobian matrix always depends on the dimension of x and the number of columns depends on the parameters. Therefore, the answer is Dimension of x (for rows). As many parameters (for column)
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8. (i)Redo: For what type of square matrices A will the algebraic system Ax = 2x have solutions other than x = 0? { Here x is an nx1 real matrix & A is an nxn real matrix.} Hints: 1) Ix = x. 2) Use GP appropriately; 3) Such matrices A must satisfy a certain relation.
Consider the quadratic form Q(x, y) = -² A = la cla 14 9 ala 10 Ila 28 9 ---/- 14 9 14 9 14 9 20 9 56 x y 9 following principal minors of the matrix A: The (leading) principal minor 1pm3 = pm3 of order 3: The leading principal minor 1pm² = pm3 of order 2: The leading principal minor 1pm₁ = pm of order 1: The principal minor pm² of order 2: The principal minor pm² of order 2: The principal minor pm of order 1: The principal minor pm of order 1: 9 Now use your answers to classify Q: (No answer given) -28 x z We want to classify this using Sylvester's Criteria (the principal minor test). Calculate the 28 y z 9 + 202² with symmetric coefficient matrix
1. a) Let x is n-vector and pand q are two constant n-vector. f(x) = ßx – q² - |l8x – q1? is f(x) linear, or affine or neither? Justify your answer. Show all calculations clearly. b) Here's a system of 512 linear equations in 512 variables x, X3. .. X512: 513 X2 + X3 + + X311 + X512i %3D 514 X1 + X3 + + X311 + X5125 1023 X1 + X2 + + X510 + X512i X1 + X2 + + X310 + X511- 1024 = Does it have a unique solution? 2. Consider the following matrix A a. Perform LU factorization. i.e Find L and U so that PA = LU. You must calculate permutation matrices- (if needed) and elementary matrices to find Land U. 0 1 3 5 1 3. Let x = (2 531), a = (4 231) be two vectors in R“. Find a matrix A where y = Ax and x is the projection of x over a. 4. Set up and solve a linear system to find vector x in R° that are simultaneously perpendicular to the vector (2,1,1)" and (2,1,3)". Can you explain and show it geometrically? 2.

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