- 9.6.a: 0, 4
- 9.6.1: In the process called ________, you are asked to find the maximum o...
- 9.6.b: 2, 3
- 9.6.2: One type of optimization strategy is called ________ ________.
- 9.6.c: 5, 0
- 9.6.3: The ________ function of a linear programming problem gives the qua...
- 9.6.d: 0, 0
- 9.6.4: The ________ of a linear programming problem determine the set of _...
- 9.6.5: The feasible solutions are ________ or ________ the boundary of the...
- 9.6.6: If a linear programming problem has a solution, it must occur at a ...
- 9.6.7: Objective function
- 9.6.8: Objective function
- 9.6.9: Objective function:
- 9.6.10: Objective function: Constraints:
- 9.6.11: Objective function:
- 9.6.12: Objective function: Constraints:
- 9.6.13: Objective function:
- 9.6.14: Objective function: Constraints:
- 9.6.15: Objective function
- 9.6.16: Objective function
- 9.6.17: Objective function
- 9.6.18: Objective function
- 9.6.19: Objective function
- 9.6.20: Objective function
- 9.6.21: z 2x y
- 9.6.22: z 5x y
- 9.6.23: z x y
- 9.6.24: z 3x y
- 9.6.25: z x 5y
- 9.6.26: z 2x 4y
- 9.6.27: z 4x 5y
- 9.6.28: z 4x y
- 9.6.29: Objective function
- 9.6.30: Objective function
- 9.6.31: Objective function
- 9.6.32: Objective function
- 9.6.33: Objective function
- 9.6.34: Objective function
- 9.6.35: OPTIMAL PROFIT A merchant plans to sell two models of MP3 players a...
- 9.6.36: OPTIMAL PROFIT A manufacturer produces two models of elliptical cro...
- 9.6.37: OPTIMAL COST An animal shelter mixes two brands of dog food. Brand ...
- 9.6.38: OPTIMAL COST A humanitarian agency can use two models of vehicles f...
- 9.6.39: OPTIMAL REVENUE An accounting firm has 780 hours of staff time and ...
- 9.6.40: OPTIMAL REVENUE The accounting firm in Exercise 39 lowers its charg...
- 9.6.41: MEDIA SELECTION A company has budgeted a maximum of $1,000,000 for ...
- 9.6.42: OPTIMAL COST According to AAA (Automobile Association of America), ...
- 9.6.43: INVESTMENT PORTFOLIO An investor has up to $250,000 to invest in tw...
- 9.6.44: INVESTMENT PORTFOLIO An investor has up to $450,000 to invest in tw...
- 9.6.45: If an objective function has a maximum value at the vertices and yo...
- 9.6.46: If an objective function has a minimum value at the vertex you can ...
- 9.6.47: When solving a linear programming problem, if the objective functio...
- 9.6.48: CAPSTONE Using the constraint region shown below, determine which o...
Solutions for Chapter 9.6: Linear Programming
Full solutions for Algebra and Trigonometry | 8th Edition
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.
Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x - x) (x - x) T is positive (semi)definite; :E is diagonal if the Xi are independent.
Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
A sequence of steps intended to approach the desired solution.
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.
Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.
A directed graph that has constants Cl, ... , Cm associated with the edges.
Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.