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Solutions for Chapter 7.1: Directed Graphs and Binary Relations; Warshalls Algorithm

Mathematical Structures for Computer Science | 7th Edition | ISBN: 9781429215107 | Authors: Judith L. Gersting

Full solutions for Mathematical Structures for Computer Science | 7th Edition

ISBN: 9781429215107

Mathematical Structures for Computer Science | 7th Edition | ISBN: 9781429215107 | Authors: Judith L. Gersting

Solutions for Chapter 7.1: Directed Graphs and Binary Relations; Warshalls Algorithm

Solutions for Chapter 7.1
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Textbook: Mathematical Structures for Computer Science
Edition: 7
Author: Judith L. Gersting
ISBN: 9781429215107

Since 46 problems in chapter 7.1: Directed Graphs and Binary Relations; Warshalls Algorithm have been answered, more than 19012 students have viewed full step-by-step solutions from this chapter. Mathematical Structures for Computer Science was written by and is associated to the ISBN: 9781429215107. Chapter 7.1: Directed Graphs and Binary Relations; Warshalls Algorithm includes 46 full step-by-step solutions. This textbook survival guide was created for the textbook: Mathematical Structures for Computer Science, edition: 7. This expansive textbook survival guide covers the following chapters and their solutions.

Key Math Terms and definitions covered in this textbook
  • Affine transformation

    Tv = Av + Vo = linear transformation plus shift.

  • Block matrix.

    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.

  • Column space C (A) =

    space of all combinations of the columns of 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

  • Diagonalization

    A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.

  • Dimension of vector space

    dim(V) = number of vectors in any basis for V.

  • Eigenvalue A and eigenvector x.

    Ax = AX with x#-O so det(A - AI) = o.

  • Exponential eAt = I + At + (At)2 12! + ...

    has derivative AeAt; eAt u(O) solves u' = Au.

  • Fibonacci numbers

    0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

  • 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.

  • Left nullspace N (AT).

    Nullspace of AT = "left nullspace" of A because y T A = OT.

  • 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).

  • Rank one matrix A = uvT f=. O.

    Column and row spaces = lines cu and cv.

  • Schur complement S, D - C A -} B.

    Appears in block elimination on [~ g ].

  • Simplex method for linear programming.

    The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

  • Toeplitz matrix.

    Constant down each diagonal = time-invariant (shift-invariant) filter.

  • Unitary matrix UH = U T = U-I.

    Orthonormal columns (complex analog of Q).

  • 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.

  • Volume of box.

    The rows (or the columns) of A generate a box with volume I det(A) I.

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