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Solutions for Chapter 15: Graphs, Charts, and Numbers

Full solutions for Excursions in Modern Mathematics | 8th Edition

ISBN: 9781292022048

Solutions for Chapter 15: Graphs, Charts, and Numbers

Solutions for Chapter 15
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Textbook: Excursions in Modern Mathematics
Edition: 8
Author: Peter Tannenbaum
ISBN: 9781292022048

This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Excursions in Modern Mathematics, edition: 8. Chapter 15: Graphs, Charts, and Numbers includes 80 full step-by-step solutions. Since 80 problems in chapter 15: Graphs, Charts, and Numbers have been answered, more than 13545 students have viewed full step-by-step solutions from this chapter. Excursions in Modern Mathematics was written by and is associated to the ISBN: 9781292022048.

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

    Tv = Av + Vo = linear transformation plus shift.

  • Augmented matrix [A b].

    Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

  • Characteristic equation det(A - AI) = O.

    The n roots are the eigenvalues of A.

  • Eigenvalue A and eigenvector x.

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

  • Elimination.

    A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

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

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

  • Free columns of A.

    Columns without pivots; these are combinations of earlier columns.

  • Gauss-Jordan method.

    Invert A by row operations on [A I] to reach [I A-I].

  • 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 .

  • Lucas numbers

    Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

  • Network.

    A directed graph that has constants Cl, ... , Cm associated with the edges.

  • Normal matrix.

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

  • Outer product uv T

    = column times row = rank one matrix.

  • Projection matrix P onto subspace S.

    Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.

  • Projection p = a(aTblaTa) onto the line through a.

    P = aaT laTa has rank l.

  • Similar matrices A and B.

    Every B = M-I AM has the same eigenvalues as A.

  • Trace of A

    = sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

  • Triangle inequality II u + v II < II u II + II v II.

    For matrix norms II A + B II < II A II + II B II·

  • Unitary matrix UH = U T = U-I.

    Orthonormal columns (complex analog of Q).

  • Vector addition.

    v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

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