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Textbooks / Math / Mathematics: A Discrete Introduction 3

Mathematics: A Discrete Introduction 3rd Edition - Solutions by Chapter

Full solutions for Mathematics: A Discrete Introduction | 3rd Edition

ISBN: 9780840049421

Mathematics: A Discrete Introduction | 3rd Edition - Solutions by Chapter

Solutions by Chapter
4 5 0 364 Reviews
Textbook: Mathematics: A Discrete Introduction
Edition: 3
Author: Edward A. Scheinerman
ISBN: 9780840049421

Since problems from 69 chapters in Mathematics: A Discrete Introduction have been answered, more than 14506 students have viewed full step-by-step answer. This expansive textbook survival guide covers the following chapters: 69. The full step-by-step solution to problem in Mathematics: A Discrete Introduction were answered by , our top Math solution expert on 03/15/18, 06:06PM. This textbook survival guide was created for the textbook: Mathematics: A Discrete Introduction, edition: 3. Mathematics: A Discrete Introduction was written by and is associated to the ISBN: 9780840049421.

Key Math Terms and definitions covered in this textbook
  • Change of basis matrix M.

    The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

  • Full column rank r = n.

    Independent columns, N(A) = {O}, no free variables.

  • Fundamental Theorem.

    The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

  • Gauss-Jordan method.

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

  • Graph G.

    Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.

  • Hessenberg matrix H.

    Triangular matrix with one extra nonzero adjacent diagonal.

  • Inverse matrix A-I.

    Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.

  • Kirchhoff's Laws.

    Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

  • Linear combination cv + d w or L C jV j.

    Vector addition and scalar multiplication.

  • Minimal polynomial of A.

    The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).

  • Pascal matrix

    Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

  • Polar decomposition A = Q H.

    Orthogonal Q times positive (semi)definite H.

  • Rank one matrix A = uvT f=. O.

    Column and row spaces = lines cu and cv.

  • Reduced row echelon form R = rref(A).

    Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

  • 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!

  • Spectrum of A = the set of eigenvalues {A I, ... , An}.

    Spectral radius = max of IAi I.

  • Toeplitz matrix.

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

  • Tridiagonal matrix T: tij = 0 if Ii - j I > 1.

    T- 1 has rank 1 above and below diagonal.

  • Vector v in Rn.

    Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.