 12.8.1: A linear programming problem requires that a linear expression, cal...
 12.8.2: True or False If a linear programming problem has a solution, it is...
 12.8.3: In 38, find the maximum and minimum value of the given objective fu...
 12.8.4: In 38, find the maximum and minimum value of the given objective fu...
 12.8.5: In 38, find the maximum and minimum value of the given objective fu...
 12.8.6: In 38, find the maximum and minimum value of the given objective fu...
 12.8.7: In 38, find the maximum and minimum value of the given objective fu...
 12.8.8: In 38, find the maximum and minimum value of the given objective fu...
 12.8.9: In 918,solve each linear programming problem.
 12.8.10: In 918,solve each linear programming problem.
 12.8.11: In 918,solve each linear programming problem.
 12.8.12: In 918,solve each linear programming problem.
 12.8.13: In 918,solve each linear programming problem.
 12.8.14: In 918,solve each linear programming problem.
 12.8.15: In 918,solve each linear programming problem.
 12.8.16: In 918,solve each linear programming problem.
 12.8.17: In 918,solve each linear programming problem.
 12.8.18: In 918,solve each linear programming problem.
 12.8.19: Maximizing Profit A manufacturer of skis produces two types: downhi...
 12.8.20: Farm Management A farmer has 70 acres of land available for plantin...
 12.8.21: Banquet Seating A banquet hall offers two types of tables for rent:...
 12.8.22: Spring Break The student activities department of a community colle...
 12.8.23: Return on Investment An investment broker is instructed by her clie...
 12.8.24: Production Scheduling In a factory, machine 1 produces 8inch (in.)...
 12.8.25: Managing a Meat Market A meat market combines ground beef and groun...
 12.8.26: Ice Cream The Mom and Pop Ice Cream Company makes two kinds of choc...
 12.8.27: Maximizing Profit on Ice Skates A factory manufactures two kinds of...
 12.8.28: Financial Planning A retired couple has up to $50,000 to place in f...
 12.8.29: Product Design An entrepreneur is having a design group produce at ...
 12.8.30: Animal Nutrition Kevins dog Amadeus likes two kinds of canned dog f...
 12.8.31: Airline Revenue An airline has two classes of service: first class ...
 12.8.32: Explain in your own words what a linear programming problem is and ...
Solutions for Chapter 12.8: Algebra and Trigonometry 9th Edition
Full solutions for Algebra and Trigonometry  9th Edition
ISBN: 9780321716569
Solutions for Chapter 12.8
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 32 problems in chapter 12.8 have been answered, more than 96689 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Algebra and Trigonometry, edition: 9. Algebra and Trigonometry was written by and is associated to the ISBN: 9780321716569. Chapter 12.8 includes 32 full stepbystep solutions.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

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

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. 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.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

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.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

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)jI and det V = product of (Xk  Xi) for k > i.