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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

ISBN: 9780470876398

Solutions for Chapter 5.4: The Simplex Method for Problems Not in Standard Form

Solutions for Chapter 5.4
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Textbook: Finite Mathematics, Binder Ready Version: An Applied Approach
Edition: 11
Author: Michael Sullivan
ISBN: 9780470876398

Finite Mathematics, Binder Ready Version: An Applied Approach was written by and is associated to the ISBN: 9780470876398. This textbook survival guide was created for the textbook: Finite Mathematics, Binder Ready Version: An Applied Approach, edition: 11. Since 21 problems in chapter 5.4: The Simplex Method for Problems Not in Standard Form have been answered, more than 16681 students have viewed full step-by-step solutions from this chapter. Chapter 5.4: The Simplex Method for Problems Not in Standard Form includes 21 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions.

Key Math Terms and definitions covered in this textbook
  • Associative Law (AB)C = A(BC).

    Parentheses can be removed to leave ABC.

  • Cholesky factorization

    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)ยท(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 .

  • Multiplier eij.

    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.

  • Vector addition.

    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.

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