SECTION 9.5
THE SIMPLEX METHOD: MIXED CONSTRAINTS
557
9.5
THE SIMPLEX METHOD: MIXED CONSTRAINTS
In Sections 9.3 and 9.4, you looked at linear programming problems that occurred in standard form. The constraints for the maximization problems all involved i
546
CHAPTER 9
LINEAR PROGRAMMING
9.4 THE SIMPLEX METHOD: MINIMIZATION
In Section 9.3, the simplex method was applied only to linear programming problems in standard form where the objective function was to be maximized. In this section, this procedure wil
Answer Key
A9
ANSWER KEY
Chapter 1
Section 1.1 (page 11)
1. Linear 7. x 2t y t 11. x1 x2 17.
4 3 2 1 -4 -2 -2 -4
25.
6 4 -2 -4 -6 -8 -10 -12
y
27.
2x - y = 12
x 1 2 3 6 7 8 6 4 2
y
0.05x - 0.03y = 0.07
x
3. Not linear 9. x y z 13. x y z
y
3 2 3 2
5. Not l
522
CHAPTER 9
LINEAR PROGRAMMING
9.2 LINEAR PROGRAMMING INVOLVING TWO VARIABLES
Many applications in business and economics involve a process called optimization, in which it is required to find the minimum cost, the maximum profit, or the minimum use of
530
CHAPTER 9
LINEAR PROGRAMMING
9.3 THE SIMPLEX METHOD: MAXIMIZATION
For linear programming problems involving two variables, the graphical solution method introduced in Section 9.2 is convenient. However, for problems involving more than two variables o
CHAPTER 9
REVIEW EXERCISES
567
CHAPTER 9
REVIEW EXERCISES
8. A warehouse operator has 24,000 square meters of floor space in which to store two products. Each unit of product I requires 20 square meters of floor space and costs $12 per day to store. Each
10.1 Gaussian Elimination with Partial Pivoting 10.2 Iterative Methods for Solving Linear Systems 10.3 Power Method for Approximating Eigenvalues 10.4 Applications of Numerical Methods
NUMERICAL METHODS
C
arl Gustav Jacob Jacobi was the second son of a su
578
CHAPTER 10
NUMERICAL METHODS
10.2 ITERATIVE METHODS FOR SOLVING LINEAR SYSTEMS
As a numerical technique, Gaussian elimination is rather unusual because it is direct. That is, a solution is obtained after a single application of Gaussian elimination. O
586
CHAPTER 10
NUMERICAL METHODS
10.3 POWER METHOD FOR APPROXIMATING EIGENVALUES
In Chapter 7 you saw that the eigenvalues of an n characteristic equation
n
n matrix A are obtained by solving its
cn
1
n 1
cn
2
n 2
.
c0
0.
For large values of n, polynomial
594
CHAPTER 10
NUMERICAL METHODS
10.4 APPLICATIONS OF NUMERICAL METHODS Applications of Gaussian Elimination with Pivoting
In Section 2.5 you used least squares regression analysis to find linear mathematical models that best fit a set of n points in the
INDEX
A
Abstraction, 191 Addition of matrices, 48 of vectors, 180, 182, 191 Additive identity of a matrix, 62 of a vector, 186, 191 properties of, 182, 185, 186 Additive inverse of a matrix, 62 of a vector, 186, 191 properties of, 182, 185, 186 Adjoining
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9.1 Systems of Linear Inequalities 9.2 Linear Programming Involving Two Variables 9.3 The Simplex Method: Maximization 9.4 The Simplex Method: Minimization 9.5 The Simplex Method: Mixed Constraints
LINEAR PROGRAMMING
ohn von Neumann was born in Budapest,
CHAPTER 8
REVIEW EXERCISES
511
CHAPTER 8
REVIEW EXERCISES
23. (1 2i (1 2i 3 3i
1
In Exercises 16, perform the given operation. 1. Find u z : u 2 4i, z 4i 2. Find u z : u 4, z 8i 3. Find uz : u 4 2i, z 4 2i 4. Find uz : u 2i, z 1 2i u 5. Find : u 6 2i, z 3
500
CHAPTER 8
COMPLEX VECTOR SPACES
8.5 UNITARY AND HERMITIAN MATRICES
Problems involving diagonalization of complex matrices and the associated eigenvalue problems require the concept of unitary and Hermitian matrices. These matrices roughly correspond t
CHAPTER 1
MATLAB EXERCISES
11
CHAPTER 1
MATLAB EXERCISES
1. Consider the linear system of Example 7 in Section 1.2. x 2y 3z 9 x 3y 4 2x 5y 5z 17 (a) Use the MATLAB command rref to solve the system. (b) Let A be the coefficient matrix of the system, and B
2
2.1 Operations with Matrices 2.2 Properties of Matrix Operations 2.3 The Inverse of a Matrix 2.4 Elementary Matrices 2.5 Applications of Matrix Operations
Matrices
CHAPTER OBJECTIVES Write a system of linear equations represented by a matrix, as well as
CHAPTER 3
MATLAB EXERCISES
31
CHAPTER 3
MATLAB EXERCISES
1. Use MATLAB to calculate the determinants of the following matrices. 5 6 7 2 3 0 1 2 (a) (b) 6 9 4 0 3 (c) pascal 4 (d) hilb 8 2. Let 1 4 . A 2 1 Use the MATLAB determinant command det to compute
CHAPTER 4
MATLAB EXERCISES
41
CHAPTER 4
MATLAB EXERCISES
1. Let u1 1, 1, 2, 2 , u2 2, 3, 5, 6 , and u3 2, 1, 3, 6 . Use MATLAB to write (if possible) the vector v as a linear combination of the vectors u1, u2, and u3. (a) v 0, 5, 3, 0 1, 6, 1, 4 (b) v 2.
CHAPTER 5
MATLAB EXERCISES
51
CHAPTER 5
MATLAB EXERCISES
1. Use the MATLAB command norm(v) to find
(a) the length of the vector v 0, 2, 1, 4, 2 . (b) a unit vector in the direction of v 3, 2, 4, 5, 0, 1 . (c) the distance between the vectors u 0, 2, 2, 3
CHAPTER 6
MATLAB EXERCISES
6 1
CHAPTER 6
MATLAB EXERCISES
1. Find the kernel and range of the linear transformation given by T x 1 1 1 2 1 9 5 13 0 1 1 0 2 10 6 14 0 0 1 2 3 11 7 15 4 12 8 16 1 0 1 0 1 2 3 2 4 Ax for these matrices A. 2 2 5
(a) A
(b) A
(c
CHAPTER 7
MATLAB EXERCISES
71
CHAPTER 7
MATLAB EXERCISES
1. The MATLAB command poly(A) produces the coefficients of the characteristic polynomial of the square matrix A, beginning with the highest degree term. Find the characteristic polynomial of the fol
CHAPTER 8
MATLAB EXERCISES
81
CHAPTER 8
MATLAB EXERCISES
1. MATLAB handles complex numbers and matrices in much the same way as real ones. The imaginary unit i 1 is a built-in constant. For instance, the complex number 2 3i would be represented as 2 3 i i
8.1 Complex Numbers 8.2 Conjugates and Division of Complex Numbers 8.3 Polar Form and DeMoivre's Theorem 8.4 Complex Vector Spaces and Inner Products 8.5 Unitary and Hermitian Matrices
COMPLEX VECTOR SPACES
C
harles Hermite was born on Christmas Eve in Di
SECTION 8.2
CONJUGATES AND DIVISION OF COMPLEX NUMBERS
477
8.2 CONJUGATES AND DIVISION OF COMPLEX NUMBERS
In Section 8.1, it was mentioned that the complex zeros of a polynomial with real coeffix 2 6x 13 cients occur in conjugate pairs. For instance, you
SECTION 8.3
POLAR FORM AND DEMOIVRE'S THEOREM
483
8.3 POLAR FORM AND DEMOIVRE'S THEOREM
Figure 8.6
Imaginary axis
(a, b) r b
At this point you can add, subtract, multiply, and divide complex numbers. However, there is still one basic procedure that is mis
SECTION 8.4
COMPLEX VECTOR SPACES AND INNER PRODUCTS
493
8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS
All the vector spaces you have studied thus far in the text are real vector spaces because the scalars are real numbers. A complex vector space is one in