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Solutions for Chapter 1.SE: Discrete Mathematics and Its Applications 7th Edition

Discrete Mathematics and Its Applications | 7th Edition | ISBN: 9780073383095 | Authors: Kenneth Rosen

Full solutions for Discrete Mathematics and Its Applications | 7th Edition

ISBN: 9780073383095

Discrete Mathematics and Its Applications | 7th Edition | ISBN: 9780073383095 | Authors: Kenneth Rosen

Solutions for Chapter 1.SE

Solutions for Chapter 1.SE
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Textbook: Discrete Mathematics and Its Applications
Edition: 7
Author: Kenneth Rosen
ISBN: 9780073383095

This expansive textbook survival guide covers the following chapters and their solutions. Chapter 1.SE includes 46 full step-by-step solutions. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095. Since 46 problems in chapter 1.SE have been answered, more than 114988 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7.

Key Math Terms and definitions covered in this textbook
  • Adjacency matrix of a graph.

    Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

  • Back substitution.

    Upper triangular systems are solved in reverse order Xn to Xl.

  • 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.)

  • Commuting matrices AB = BA.

    If diagonalizable, they share n eigenvectors.

  • Cyclic shift

    S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

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

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

  • Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).

    Use AT for complex A.

  • Fourier matrix F.

    Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

  • 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!).

  • Gram-Schmidt orthogonalization A = QR.

    Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

  • 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.

  • Kronecker product (tensor product) A ® B.

    Blocks aij B, eigenvalues Ap(A)Aq(B).

  • 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 .

  • Multiplication Ax

    = Xl (column 1) + ... + xn(column n) = combination of columns.

  • 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.

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

    P = aaT laTa has rank l.

  • Semidefinite matrix A.

    (Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

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

    Spectral radius = max of IAi I.

  • Sum V + W of subs paces.

    Space of all (v in V) + (w in W). Direct sum: V n W = to}.

  • Symmetric factorizations A = LDLT and A = QAQT.

    Signs in A = signs in D.

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