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Introduction to Linear Algebra | 4th Edition | ISBN: 9780980232714 | Authors: Gilbert Strang
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Author: Gilbert Strang
Publisher: Wellesley Cambridge Press
ISBN: 9780980232714
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Chapter 8 Problems

Chapter: 8 Problem: 1

Introduction to Linear Algebra 4
Draw the region in the xy plane where x + 2y = 6 and x > and y > 0. Which point in this "feasible set" minimizes the cost c = x + 3y? Which point gives maximum cost? Those points are at comers.

Chapter: 8 Problem: 2

Introduction to Linear Algebra 4
Draw the region in the xy plane where x + 2y < 6, 2x + y < 6, x > 0, y > 0. It has four comers. Which comer minimizes the cost c = 2x - y?

Chapter: 8 Problem: 3

Introduction to Linear Algebra 4
What are the comers of the set Xl + 2X2 - X3 = 4 with Xl, x2, x3 all > o? Show that the cost Xl + 2X3 can be very negative in this feasible set. This is an example of unbounded cost: no minimum.

Chapter: 8 Problem: 4

Introduction to Linear Algebra 4
Start at x = (0,0,2) where the machine solves all four problems for $16. Move to x = (0,1, ) to find the reduced cost r (the savings per hour) for work by the student. Find r for the Ph.D. by moving to x = (I, 0, ) with 1 hour of Ph.D. work.

Chapter: 8 Problem: 5

Introduction to Linear Algebra 4
Start Example 1 from the Ph.D. comer (4,0,0) with c changed to [5 3 7]. Show that r is better for the machine even when the total cost is lower for the student. The simplex method takes two steps, first to the machine and then to the student for x *.

Chapter: 8 Problem: 6

Introduction to Linear Algebra 4
Choose a different cost vector c so the Ph.D. gets the job. Rewrite the dual problem (maximum income to the cheater).

Chapter: 8 Problem: 7

Introduction to Linear Algebra 4
A six-problem homework on which the Ph.D. is fastest gives a second constraint 2Xl + X2 + X3 = .6. Then x = (2,2,0) shows two hours of work by Ph.D. and student on each ho~ework. Does this x minimize the cost c T x with c = (5,3,8) ?

Chapter: 8 Problem: 8

Introduction to Linear Algebra 4
These two problems are also dual. Prove weak duality, that always y T b < C T x: Primal problem Minimize c T x with Ax > b and x > O. Dual problem Maximize y T b with AT y < c and y ~ O.

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