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There are two files in which one is written homework and the other one requires Matlab. Please do the questions the way before. Thank you!Math 171A: Linear Programming
Instructor: Philip E. Gill © 2021 (Not to be Reposted)
Winter Quarter 2021
Matlab Assignment #3
Due Wednesday February 10, 2021
The starred exercises require the use of Matlab. Remember that it is necessary to do all
the Matlab assignments to obtain credit for the class.
Exercise 3.1.∗
(a) Define range(A), null(AT ), range(AT ), null(A) and rank(A) for a general matrix A.
(b) Consider the matrix

−1
 1

A=
2
−3
1
2
3
0

2
1
.
1
3
(i) Use the Matlab command rank to find rank(A).
(ii) Find the dimensions of the subspaces range(A), null(AT ), range(AT ) and null(A).
(iii) Use the Matlab command null to find a basis for null(AT ).
Exercise 3.2.∗ Consider the matrix A and vector y given by

A=

−1
0
−9
0
−3
−6
1
0
2
−3
−2
0
−6
4
−1
0
−2
−1
−4
4
2
−2
−2
−8
1
−7
4
−1

and y = 

−7
−3
1
−1

.

(a) Find the rank of A. Use the Matlab command null to find a basis for null(A).
(b) Find the unique vectors yR ∈ range(AT ) and yN ∈ null(A) such that y = yR + yN .
Check that your computed yN and yR satisfy yRT yN = 0.
Exercise 3.3.∗ Consider the equality-constrained linear program: minimize cTx subject to
Ax = b. Consider two problems defined with data:
 
4
−6

 
−1
5
0
1
1
4

, and c = 
(a) A =
, b=
 1.
3 −1
1
4
2
−5
 3
1
2
Mathematics 171A

1
 2
 

c=
−4.
 0
−1

−3
(b) A = −3
−6

−2 −1 −2 −6
3 −6
0 −3,
−6
0 −1
0
 
−6
b =  0,
−5
and
For each of these problems, find a feasible point x̄. Find if x̄ is optimal. If x̄ is not optimal,
find unique vectors cN ∈ null(A) and cR ∈ range(AT ) such that c = cR + cN . Hence find a
feasible direction p such that `(x̄ + αp) < `(x̄) for all positive α. Math 171A: Linear Programming Instructor: Philip E. Gill © 2021 (Not to be Reposted) Winter Quarter 2021 Homework Assignment #3 Due Wednesday February 10, 2021 Exercise 3.1. Given a nonzero matrix A every nonzero vector b may be written as b = bR + bN , where bR ∈ range(A) and bN ∈ null(AT ). (a) Show that bR and bN are unique. (b) Show that bTR bN = 0. (c) If bR and bN are both nonzero, show that they are linearly independent. Exercise 3.2. Consider the matrix 1 A= 2 −1 −2 3 6 2 −1 . 4 −2 (a) Give a general expression for all vectors in the range of A. What is the rank of A? (b) Using part (a), give a nonzero vector b for which Ax = b is compatible. (c) Give a vector b such that Ax = b is not compatible, and explain how you found b. Exercise 3.3. Suppose that the constant vector c is such that cTp ≥ 0 for all p such that Ap = 0. Show that this implies that cTp = 0 for all p such that Ap = 0. Exercise 3.4. Consider the equality-constrained linear program: minimize cTx subject to Ax = b. If this problem has a bounded minimum `∗ , show the following: (i) `∗ = cTx̄ = cTxb for any two feasible points x̄ and xb; b for any two sets of Lagrange multipliers λ̄ and λ. b (ii) `∗ = bT λ̄ = bT λ Exercise 3.5. Consider the equality-constrained linear program: minimize cTx subject to Ax = b. Show that if A has full column rank and b ∈ range(A), then this problem always has a unique bounded solution, regardless of the definition of c. Do Lagrange multipliers exist? If they exist, are they unique? Purchase answer to see full attachment

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