- 5.4.1: True or False In using the Alternate Pivoting Strategy, some of the...
- 5.4.2: A minimum problem that is not in standard form and whose objective ...
- 5.4.3: True or False In solving a maximum problem not in standard form, th...
- 5.4.4: True or False If a constraint is an equality, it can be ignored.
- 5.4.5: In 510, solve each maximum problem not in standard form.P = 3x1 + 4x2
- 5.4.6: In 510, solve each maximum problem not in standard form.P = 5x1 + 2x2
- 5.4.7: In 510, solve each maximum problem not in standard form.P = 3x1 + 2...
- 5.4.8: In 510, solve each maximum problem not in standard form.P = 3x1 + 2...
- 5.4.9: In 510, solve each maximum problem not in standard form.P = 3x1 + 2x2
- 5.4.10: In 510, solve each maximum problem not in standard form.P = 45x1 + ...
- 5.4.11: In 1112, solve each minimum problem not in standard from.z = 6x1 + ...
- 5.4.12: In 1112, solve each minimum problem not in standard from.z = 2x1 + ...
- 5.4.13: Shipping Private Motors, Inc., has two plants, M1 and M2, which man...
- 5.4.14: . Shipping Schedule A cell phone manufacturer must fill orders from...
- 5.4.15: . Shipping Schedule A GPS systems manufacturer must fill orders fro...
- 5.4.16: Mixture Minimize the cost of preparing the following mixture, which...
- 5.4.17: Advertising A local appliance store has decided on an advertising c...
- 5.4.18: Minimizing Materials Quality Oak Tables, Inc., has an individual wh...
- 5.4.19: Nutrition The serving sizes and nutritional content per serving of ...
- 5.4.20: Maximizing Salespeople at a Trade Show Suppose that a computer comp...
- 5.4.21: Pension Funds A pension fund has decided to invest $50,000 in the f...
Solutions for Chapter 5.4: The Simplex Method for Problems Not in Standard Form
Full solutions for Finite Mathematics, Binder Ready Version: An Applied Approach | 11th Edition
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
A = CTC = (L.J]))(L.J]))T for positive definite A.
Dimension of vector space
dim(V) = number of vectors in any basis for V.
Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
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 variable Xi.
Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
Incidence matrix of a directed graph.
The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .
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.
Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).
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.
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.
Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.