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

#### Why Choose Us

- 100% non-plagiarized Papers
- 24/7 /365 Service Available
- Affordable Prices
- Any Paper, Urgency, and Subject
- Will complete your papers in 6 hours
- On-time Delivery
- Money-back and Privacy guarantees
- Unlimited Amendments upon request
- Satisfaction guarantee

#### How it Works

- Click on the “Place Order” tab at the top menu or “Order Now” icon at the bottom and a new page will appear with an order form to be filled.
- Fill in your paper’s requirements in the "
**PAPER DETAILS**" section. - Fill in your paper’s academic level, deadline, and the required number of pages from the drop-down menus.
- Click “
**CREATE ACCOUNT & SIGN IN**” to enter your registration details and get an account with us for record-keeping and then, click on “PROCEED TO CHECKOUT” at the bottom of the page. - From there, the payment sections will show, follow the guided payment process and your order will be available for our writing team to work on it.